17  g  r .  /  •?  n  n  ir  .s  1)^  .->  / 


ADVANCED  LABORATORY  PRACTICE 

IN 

ELECTRICITY  AND  MAGNETISM 


'3/ill  Book  (a  7m 


PUBLISHERS     OF     BOOKS     F  O  R_/ 

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Industrial  Engineer 


ADVANCED 
LABOEATORY  PRACTICE 

IN 

ELECTRICITY 

'     .  .  "       -  AND 

MAGNETISM 


BY 


EARLE  MELVIN  TERRY,  PH.D. 

M 

ASSOCIATE   PROFESSOR   OP   PHYSICS,    UNIVERSITY    OP    WISCONSIN 


FIRST  EDITION 


McGRAW-HILL  BOOK  COMPANY,  INC. 
NEW  YORK:  370  SEVENTH  AVENUE 

LONDON:  6  &  8  BOQVERIE  ST.,  E.  C.  4 
1922 


3 


Engineering 


COPYRIGHT,  1922,  BY  THE 
McGRAW-HiLL  BOOK  COMPANY,  INC. 


THE    MAPLE   PRESS     -     YORK   PA 


PREFACE 

In  preparing  this  book,  the 'author  has  had  in  mind  particularly 
the  needs  of  those  students  whb  have  at  their  disposal  only  one 
year  to  devote  to  the  study  of  electricity  and  magnetism  in  addi- 
tion to  the  work  covered  in  an  elementary  course  in  general 
physics.  It  has  been  his  aim  to  include,  in  addition  to  the  usual 
work  in  electrical  measurements,  a  sufficient  study  of  the  dis- 
charge of  electricity  through  gases,  radio  activity,  and  thermionics 
to  enable  those  who  cannot  pursue  special  courses  to  gain  an 
idea  of  the  fundamentals  of  these  newer  branches. 

The  subject  matter  covers  the  work  given  to  third  year  students 
in  electrical  engineering  at  the  University  of  Wisconsin.  Follow- 
ing the  elementary  work  of  the  first  nine  chapters,  a  number  of 
the  complex  bridge  methods  for  precise  measurements  of  induc- 
tance and  capacitance  are  discussed,  together  with  descriptions  of 
the  various  sources  of  alternating  currents  which  have  been 
developed  in  recent  years  for  energizing  bridge  circuits.  A  dis- 
cussion of  the  more  modern  instruments  for  detecting  the  balance 
condition  of  bridges,  together  with  their  individual  merits,  has 
been  included.  This  is  preceded  by  an  elementary  study  of 
"transients,"  in  which  the  fundamental  phenomena  of  reactance, 
necessary  for  an  understanding  of  bridge  methods,  are  set  forth. 

The  electron  tube,  because  of  the  multiplicity  of  its  uses,  finds 
many  applications,  not  only  in  the  art  of  radio  communication, 
but  also  in  engineering  practice  and  in  the  general  research 
laboratory.  Considerable  space  has  been  devoted  to  this  device, 
as  well  as  to  the  fundamentals  of  the  electron  theory  and  the 
passage  of  electricity  through  gases. 

The  author  is  a  firm  believer  in  the  laboratory  method  of 
instruction,  and  each  exercise  is  preceded  by  a  discussion  of  the 
theories  involved  sufficient  to  enable  the  student  to  understand 
clearly  the  relation  of  each  experiment  to  the  general  field  in 
which  it  lies.  It  is  believed  that  with  the  material  given  in  the 
text  and  the  references  to  standard  works,  which  have  been 
included,  the  student  can  pursue  the  subject  without  the  aid  of 


48 


vi  PREFACE 

formal  lectures,  although  at  the  present  time  the  writer  is  devot- 
ing one  hour  per  week  to  a  lecture-conference.  Experience 
shows  that  the  average  student  performs  fourteen  of  these 
exercises  per  semester  and  the  topics  herewith  presented  accord- 
ingly permit  of  some  little  choice. 

In  selecting  material,  advantage  has  been  taken  not  only  of  the 
original  sources,  but  also  of  the  standard  texts  in  the  special 
fields  represented.  Being  a  collection  of  laboratory  exercises, 
this  book  makes  no  claim  to  originality  of  the  subject  matter 
included,  and  the  author  hereby  acknowledges  his  indebtedness 
to  the  many  books  and  special  articles  referred  to  in  the  foot- 
notes throughout  the  text.  He  is  indebted,  also,  to  the  Leeds- 
Northrup  Company,  J.  G.  Biddle  Company,  Queen  &  Company, 
General  Radio  Company,  Tinsley  &  Company,  and  other  manu- 
facturers of  electrical  apparatus  for  supplying  the  cuts  which  have 
been  used.  In  particular,  he  wishes  to  express  his  gratitude  to 
Dr.  H.  B.  Wahlin  and  L.  L.  Nettleton,  instructors  in  physics  at 
the  University  of  Wisconsin,  who  have  read  the  entire  manu- 
script and  made  many  vauable  suggestions  during  its 
preparation. 

E.  M.  TERRY. 
UNIVERSITY  OF  WISCONSIN, 
MADISON,  Wis. 
June,  1922. 


CONTENTS 

PAGE 
PREFACE  &  v 

•w 

CHAPTER    I. — GENERAL  DIRECTIONS — ELECTRICAL  UNITS 1 

Preparation — Connections — Keys  and  Switches — Rheostats — 
Switch  Board — Care  of  Apparatus — Notebooks — Electrostatic 
System  of  Units — Electromagnetic  System  of  Units — Practical 
System  of  Units — Ratios  of  the  Electrical  Units — Rationalized 
Practical  System  of  Units. 

CHAPTER    II.— GALVANOMETERS 17 

Thomson  Galvanometer — D' Arson  val  Galvanometer — Galvano- 
meter Sensitivity — Figure  of  Merit — Ballistic  Galvanometer — 
Constant  of  Ballistic  Galvanometer — Flux  Meter — Checking 
Devices. 

CHAPTER    III. — MEASUREMENT  OF  RESISTANCE 35 

Ohm's  Law — Specific  Resistance — Temperature  Coefficient  of 
Resistance — Wheatstone  Bridge — Measurement  of  Low  Resistance 
— Measurement  of  High  Resistance — Internal  Resistance  of  Cells — 
Battery  Test. 

CHAPTER  IV. — MEASUREMENT  OF  POTENTIAL  DIFFERENCE 55 

Description  of  Potentiometer — Direct  Reading  Potentiometer — 
Leeds  and  Northrup  Potentiometer — Wolff  Potentiometer — 
Tinsley  Potentiometer — Weston  Standard  Cell — Volt  Box. 

CHAPTER  V. — MEASUREMENT  OF  CURRENT 70 

Kelvin's  Balance — Siemen's  Electro  dynamo  meter — Ammeters  and 
Voltmeters — Adjustment  of  Ammeters  and  Voltmeters — Measure- 
ment of  Current  by  the  Potentiometer. 

CHAPTER  VI. — MEASUREMENT  OF  POWER 82 

Wattmeters,  Types  of — Compensation  of  Wattmeters — Calibra- 
tion of  an  Indicating  Wattmeter. 

CHAPTER  VII. — MEASUREMENT  OF  CAPACITANCE 86 

Condensers — Grouping  of  Condensers — Standard  Condensers — 
Comparison  of  Condensers — Flemming  and  Clinton  Commutator. 

CHAPTER  VIII.— MAGNETISM 95 

Strength  of  Pole — Strength  of  Field — Magnetic  Moment — 
Magnetic  Induction — Permeability  and  Susceptibility — 
Demagnetization  due  to  Ends  of  a  Bar  Magnet — Magnetic 
Circuit — Magnetic  Units — Magnetization  Curves — Hysteresis, 

vii 


viii  CONTENTS 

PAGE 

CHAPTER  IX.— SELF  AND  MUTUAL  INDUCTANCE 117 

General  Principles — Definition  of  Units  of  Inductance — Standards 
of  Inductance — Measurement  of  Self -inductance — Measurement 
of  Mutal  Inductance. 

CHAPTER  X. — ELEMENTARY  TRANSIENT  PHENOMENA 126 

Time  Constant — Circuit  Containing  Resistance  and  Inductance — 
Circuit  Containing  Resistance  and  Capacitance — Circuit  contain- 
ing Resistance,  Inductance  and  Capacitance — Non-oscillatory 
Discharge  of  a  Condenser — Aperiodic  Discharge  of  a  Condenser — 
Oscillatory  Discharge  of  a  Condenser — Logarithmic  Decrement — 
Harmonic  E.  M.  F.  acting  on  a  Circuit  Containing  Resistance, 
Inductance,  and  Capacitance — Vector  Diagrams — Series 
Resonance — Parallel  Resonance — Measurement  of  Inductance 
and  Capacitance  by  Resonance — Effective  Value  of  an  Alternating 
Current — Power  Consumed  by  a  Circuit  Traversed  by  an 
Alternating  Current. 

CHAPTER  XI. — SOURCES  OF  ELECTROMOTIVE  FORCE  AND  DETECTING 

DEVICES  FOR  BRIDGE  METHODS 151 

Sechometer — Wire  Interrupter — Motor  Generator — Microphone 
Hummer — Audio  Oscillator — Vreeland  Osallator — Electron  Tube 
Oscillator — Telephone  Receiver — Thermo  Galvanometer — Vibra- 
tion Galvanometer — Alternating  Current  Galvanometer. 

CHAPTER  XII. — ALTERNATING  CURRENT  BRIDGES 168 

General  Considerations — Maxwell's  Bridge — Anderson's  Modifica- 
tion of  Maxwell's  Bridge — Stroude  and  Oates'  Bridge — Trow- 
bridge's  Method — Heydweiller's  Network — Heavieside's  Bridge — 
Maxwell's  Bridge  for  Mutual  Inductance — Mutual  Inductance 
Bridge — Frequency  Bridge — Circuits  of  Variable  Impedance — 
Motional  Impedance  of  a  Telephone  Receiver — Power  Factor 
and  Capacitance  of  Condensers — Resistance  of  Electrolytes. 

CHAPTER  XIII. — CONDUCTION  OF  ELECTRICITY  THROUGH  GASES  .  .  198 
Electrons — Conductivity  of  Gases — Structure  of  the  Atom — loni- 
zation  Current — Resistance  of  a  Discharge  Tube — Phenomena  of 
the  Discharge  Tube — Theory  of  the  Discharge — Field  Strength  at 
Various  Points  in  the  Discharge — Cathode  Rays — Velocity  and 
Ratio  of  the  Charge  to  the  Mass  of  an  Electron — Radioactive 
Substances — Alpha  Rays — Beta  Rays — Gamma  Rays — Radio 
active  Transformations — lonization  by  Radio  Active  Substances. 

CHAPTER  XIV.— ELECTRON  TUBES 218 

Free  Electrons — Electron  Emission — Two  Element  Electron  Tube 
— Voltage  Saturation — Space  Charge — Characteristics  of  the 
Two  Element  Electron  Tube— Three  Element  Electron  Tube- 
Static  Characteristics — Amplification  Factor — Internal  Resistance 
of  Three  Element  Electron  Tube — Tungar  Rectifier. 


CONTENTS  ix 

PAGE 

CHAPTER  XV. — PHOTOMETER  AND  OPTICAL  PYROMETER 239 

Intensity  of  Radiation — Photometers — Lummer-Brodhun  Photo- 
meter— Stirdy  of  Incandescent  Lamps — General  Principles  of 
Radiation — Black  Body  Furnace — Distribution  of  Energy  in  the 
Spectrum — Applications  to  Pyrometry — Optical  Pyrometer. 

APPENDIX 252 

,& 
Calculation  of  Self  Inductance — Calculation  of  Capacitance. 

INDEX.  .   259 


LIST  OF  EXPERIMENTS 

PAGE 

1.  Specific  Resistance  of  Materials 42 

2.  Temperature  Coefficient  of  Resistance 45 

3.  Insulation  Resistance  by  Leakage 48 

4.  Internal  Resistance  of  Cells  by  Condenser  Method 52 

5.  Battery  Test 53 

6.  Comparison  of  E.M.F.  of  Cells  by  the  Potentiometer 66 

7.  Calibration  of  a  Voltmeter  by  the  Potentiometer  and  Volt  Box 68 

8.  Calibration  of  an  Electrodynamometer 73 

9.  Electrical  Adjustment  of  an  Ammeter  and  a  Voltmeter 77 

10.  Calibration  of  an  Ammeter  by  the  Potentiometer  and  Standard 

Resistance 80 

11.  Calibration  of  a  Wattmeter 84 

12.  Comparison  of  Capacitances  by  the  Bridge  Method 92 

13.  Capacitance  by  the  Fleming  and  Clinton  Method 93 

14.  Magnetization  Curves  by  Hopkinson's  Bar  and  Yoke 108 

15.  Magnetization  Curves  by  the  Rowland  Ring  Method 113 

16.  Hysteresis  Curves 114 

17.  Comparison  of  Self  Inductances  by  the  Bridge  Method 122 

18.  Mutual  Inductance  by  Carey-Foster's  Method 124 

19.  Measurement  of  Inductance  and  Capacitance  by  Resonance 148 

20.  Maxwell's  Bridge  for  Self  Inductance 171 

21.  Stroude  and  Gate's  Bridge  for  Self  Inductance 174 

22.  Trowbridge's  Method  for  Self  Inductance 177 

23.  Heydweiller's  Method  for  Mutual  Inductance 179 

24.  Heaviside's  Bridge  for  Self  Inductance 182 

25.  Maxwell's  Bridge  for  Mutual  Inductance 184 

26.  Comparison  of  Mutual  Inductances 185 

27.  Bridge  Method  for  Measuring  Frequency 187 

28.  Motional  Impedance  of  a  Telephone  Receiver 191 

29.  Phase  Difference  and  Capacitance  of  a  Condenser 193 

30.  Resistance  of  Electrolytes . 197 

31.  Resistance  of  a  Discharge  Tube 202 

32.  Variation  of  Field  Strength  along  the  Discharge 207 

33.  Measurement  of  —  and  Velocity  of  an  Electron 211 

34.  lonization  by  Radio  Active  Substances 216 

35.  Characteristics  of  the  Two  Element  Electron  Tube 224 

36.  Static  Characteristics  of  a  Three  Element  Electron  Tube 226 

37.  Amplification  Factor  of  a  Three  Element  Electron  Tube. 231 

38.  Plate-Filament  Resistance  of  a  Three  Element  Electron  Tube 235 

39.  Study  of  the  Tungar  Rectifier 237 

40.  Study  of  Incandescent  Lamps 243 

41.  The  Optical  Pyrometer 250 

xi 


ADVANCED  LABORATORY  PRACTICE 

m 
ELECTRICITY  AND  MAGNETISM 

CHAPTER  I 
GENERAL  DIRECTIONS— ELECTRICAL  UNITS 

1.  Preparation. — If  one  is  to  make  the  best  use  of  his  time  in 
the  laboratory,  he  must  understand  thoroughly  what  is  to  be 
done  and  then  proceed  in  a  systematic  manner  to  do  it.  This 
can  be  accomplished  only  when  preparation  for  the  task  has 
been  made  before  taking  up  the  experimental  work.  Assign- 
ments accordingly  will  be  made  one  week  in  advance,  and  the 
student  is  expected  to  enter  the  laboratory  with  the  following 
preparation : 

1.  An  understanding  of  the  theory  of  the  experiment. 

2.  A  knowledge  of  the  working  principles  of  the  instruments 
to  be  used. 

3.  A  schedule  according  to  which  the  data  are  to  be  taken. 
In  order  to  facilitate  the  work  of  the  first  few  periods,  the  follow- 
ing general  directions  should  be  carefully  read: 

2.  Connections. — A  large  portion  of  the  trouble  in  performing 
electrical  measurements  arises  from  imperfect  connections.  All 
instruments,  to  which  wires  are  to  be  attached,  are  provided  with 
binding  posts.  To  secure  good  contact,  remove  the  insulation 
about  an  inch  from  the  end  of  the  wire,  scrape  it  clean,  wrap  it 
two-thirds  around  the  binding  post,  and  then  screw  down  the 
nut.  If  the  wire  is  too  short  to  reach  between  the  points  desired, 
join  two  or  more  wires  with  connectors,  having  first  scraped  the 
ends  clean.  Never  join  wires  by  twisting  their  ends  together, 
as  connections  of  this  sort,  unless  soldered,  are  entirely  unreliable. 
Do  not  coil  wires  about  a  rod  or  a  pencil,  since  then  they  cannot 
be  used  again.  Gut  wires  to  the  proper  length,  thus  avoiding  a 
complicated  tangle  difficult  to  trace,  which,  through  leaks, 

1 


•  *«•*,  -        »  •   •  1*  « 

.*    *    \  -     - 


•  V 


£>,:  i'*;.**:V  "3a»^/fe> 


AND  MAGNETISM 

furnishes  a  source  of  constant  trouble.  Never  allow  one  wire  to 
rest  upon  another,  even  though  both  are  covered  with  insulation. 

Before  attempting  a  set-up,  make  a  rough  sketch  of  connec- 
tions, arranging  the  apparatus  in  a  compact  and  orderly  manner. 
This  will  be  of  great  service  later  in  checking  connections  and 
locating  faults.  In  many  cases,  especially  in  complicated  net 
works,  a  little  forethought  in  the  arrangement  will  save  much 
time  and  inconvenience  in  the  performance  of  the  test.  Always 
make  the  connection  with  the  source  of  current  supply  last, 
having  first  assured  yourself  as  to  the  correctness  of  the  connec- 
tions by  comparison  with  the  sketch,  or  by  consultation  with  your 
instructor.  As  a  further  precaution,  close  the  main  switch  at 
first  only  an  instant,  opening  it  at  once  to  see  if  there  are  any 
indications  of  a  short  circuit.  This  is  especially  important  where 
the  source  is  a  dynamo  or  a  storage  battery. 

3.  Keys  and  Switches. — Always  open  and  close  a  switch 
quickly,  to  avoid  burning  it  at  the  point  of  contact.  If  the 


FIG.   1. — Reversing  switch. 

circuit  includes  mercury  cups  and  connecting  links,  it  should  be 
broken  by  means  of  a  knife  switch,  not  by  removing  the  link,  as 
mercury  is  especially  likely  to  arc.  Ordinary  contact  keys  should 
be  used  only  where  a  small  current  is  to  be  carried,  and  where 
variations  in  the  resistance  of  the  circuit  introduce  no  serious 
error;  as,  for  example,  the  galvanometer  circuit  of  a  Wheatstone 
bridge. 

The  device  generally  employed  for  reversing  the  current 
through  any  portion  of  a  circuit  is  the  PohPs  commutator,  which 
consists  of  a  double  pole  double  throw  switch  with  two  cross 
wires,  as  shown  in  Fig.  1.  It  will  be  seen  that  when  the  switch  is 
closed,  as  shown  by  the  heavy  lines,  the  current  through  R  flows 
upwards,  but  is  reversed  when  the  switch  is  thrown  towards  the 
right.  Since  the  cross  wires  in  such  a  commutator  are  frequently 


ELECTRICAL  UNITS  3 

placed  under  the  block,  a  double  pole  double  throw  switch  should 
be  examined  carefully  before  it  is  connected  in  circuit,  as  the 
cross  wires  of  the  commutator  may  produce  short  circuits, 
resulting  in  serious  injury  to  the  apparatus. 

4.  Rheostats. — A  rheostat  is  a  variable  resistance  capable  of 
carrying  considerable  current.  Jt  is  used  primarily  as  a  control- 
ling device  and  its  value,  in  general,  need  not  be  accurately 
known.  When  connected  in  series  with  a  source  of  electrical 


FIG.  2. — Rheostat  with  fixed  steps. 


power,  the  current  supplied  to  any  circuit  may  be  varied  and 
brought  to  any  desired  value,  within  certain  definite  limits,  by 
changing  the  resistance  of  the  rheostat.  Since  the  energy  con- 
sumed by  a  rheostat  always  appears  in  the  form  of  heat,  the 
current  carrying  capacity  for  a  given  resistance  depends  upon 
the  provision  made  for  dissipating  heat  either  by  conduction, 
convection,  or  radiation. 

Many  different  forms  of  rheostats  are  in  use  and  only  a  few  of 
the  more  common  types  will  be  mentioned  here.  Figure  2 
illustrates  one  that  is  frequently  used  for  controlling  relatively 
large  currents.  It  consists  of  a  number  of  copper  lugs  between 
which  are  connected  units  of  high  resistance  metal  in  the  form 
of  a  thin  ribbon  to  give  as  much  heat  iradating  surface  as  possible. 
They  are  bent  back  and  forth  in  a  zig-zag  shape  and  embedded 
in  sand.  With  this  arrangement  the  resistance  varies  by  steps. 
By  connecting  two  in  series,  one  having  large  and  the  other  small 
steps,  a  fairly  smooth  variation  in  current  may  be  obtained. 


4  ELECTRICITY  AND  MAGNETISM 

Another  common  form  of  rheostat  is  shown  in  Fig.  3.  A  high 
resistance  wire  is  wound  on  an  insulating  tube.  Binding  posts 
are  connected  to  each  end  of  the  wire,  and  as  the  sliding  contact  is 
moved  along,  the  resistance  between  it  and  one  end  of  the  wire 
changes  from  zero  to  a  maximum.  Wires  of  various  sizes  are 
frequently  wound  on  the  same  tube  thus  giving  two  or  more 


FIG.  3. — Tube  rheostat. 

ranges  for  one  instrument.  For  carrying  large  currents,  the  ends 
of  the  tube  are  closed  and  cooling  water  is  passed  through. 
Such  rheostats  are  generally  wound  with  a  ribbon  to  improve  the 
thermal  contact  between  the  wire  and  tube.  The  figure  shows 
a  high  resistance  instrument,  which  may  also  be  used  as  a  poten- 
tial divider.  If  the  E.M.F.  to  be  divided  is  connected  across  the 
end  binding  posts,  any  desired  fraction  of  this  voltage  may  be 


FIG.  4. — Carbon  compression  rheostat. 

obtained  by  " picking  off"  between  one  end  and  the  slider,  and 
moving  the  latter  back  and  forth. 

Another  device  for  controlling  current  makes  use  of  the  fact 
that  the  resistance  between  two  carbon  surfaces  varies  with  the 
pressure.  An  instrument  of  this  sort  is  shown  in  Fig.  4.  It 
consists  of  series  of  rectangular  carbon  plates  placed  in  a  trough 
and  arranged  in  such  a  way  that  they  can  be  subjected  to  variable 
pressures  by  means  of  an  adjusting  screw.  Rheostats  of  this 


ELECTRICAL  UNITS  5 

type  are  useful  where  low  voltage  currents  are  to  be  controlled. 
They  have  the  disadvantage  of  requiring  frequent  readjustment 
since  the  tension  changes  with  variations  of  the  temperature  of 
both  the  carbon  plates  and  metal  parts. 

5.  Switch  Board. — A  switch  board  is  a  necessary  adjunct  to 
any   electrical   laboratory   and  is  used  to  distribute  electrical 
power  of  different  types  and  voltage  to  the  various  working  cir- 
cuits of  the  laboratory,  and  to  connect  the  different  circuits  with 
one  another.     It  consists  of  a  panel  of  insulating  material,  usu- 
ally marble,  on  which  is  mounted  a  series  of  pairs  of  sockets. 
The  various  laboratory  and  power  circuits  are  joined  to  these 
sockets  on  the  back  of  the  board  and  connections  between  them 
are  made  at  the  front  by  means  of  flexible  connectors,  often  called 
"jumpers,"  to  the  ends  of  which  are  attached  plugs  which  fit 
snugly  into  the  sockets.     Power  circuits  are  distinguished  by  the 
word  "  Volts."     With  the  exception  of  the  A.C.  circuits,  each 
terminal  is  labeled  plus  or  minus.     If,  for  example,   10  volts 
are  desired  on  circuit  91,  connect  the  positive  of  a  10  volt  set 
with  the  positive  of  91,  and  similarly  for  the  negative,  when  the 
polarity  at  the  laboratory  end  will  be  found  as  indicated.     If 
some  voltage  is  desired,  e.g.,  16  volts,  for  which  there  is  no  separate 
set,  connect  two  or  more  sets  in  series,  joining  plus  to  minus  as 
though  connecting  cells  on  a  table;  then,  considering  the  group 
as  a  single  set,  connect  to  the  laboratory  terminals  as  above.     If 
a  current  larger  than  the  normal  rated  capacity  of  the  storage 
battery  is  desired,  use  the  dynamo  or  connect  in  parallel  two  or 
more  sets  of  equal  voltage.     To  do  the  latter,  join  all  the  positive 
terminals,  similarly  the  negatives,  and  then  connect  to  the  labo- 
ratory terminals  as  above.     Before  making  switch  board  connec- 
tions, be  sure  that  the  circuit  switch  in  the  laboratory  is  open. 
Connect  the  "dead"  ends  first,  and,  before  pushing  in  the  final 
plug  closing  the  circuit,  tap  it  cautiously  against  the  socket, 
quickly  withdrawing  it.     If  a  spark  is  seen,  some  error  in  connec- 
tion has  been  made  which  must  be  located  before  the  circuit  is 
closed.     Never  connect  in  parallel  on  a  battery  with  some  one 
else  without  first  obtaining  his  permission. 

6.  Care  of  Apparatus. — Electrical  apparatus  is  delicate  and 
expensive,  and  it  is  necessary  to  proceed  with  the  utmost  caution. 
If  an  instrument  is  provided  with  a  shunt,  use  the  smallest 
resistance  first;  or,  if  protected  by  a  series  resistance,  use  the 
largest  value  first  decreasing  it  until  the  desired  value  has  been 


6  ELECTRICITY  AND  MAGNETISM 

reached.  If  an  instrument  fails  to  work,  do  not  replace  it  in 
the  case  and  get  another,  but  report  at  once  to  the  instructor. 
Resistance  boxes  are  most  frequently  injured  by  carrying  too 
large  currents.  Before  closing  the  main  switch,  look  over  the 
connections  and  make  a  rough  calculation  of  the  current  that  will 
flow  in  each  box.  In  no  case  should  the  power  consumed  by  a 
single  coil,  given  by  PR,  be  more  than  four  watts.  Plugs  should 
be  seated  by  a  gentle  pressure,  accompanied  by  a  twisting  motion, 
heavy  pressure  being  unnecessary. 

Never  move  galvanometers  from  one  place  to  another  without 
first  making  sure  that  the  weight  of  the  moving  system  has  been 
removed  from  the  suspension  by  means  of  the  arrestment  which 
is  always  provided.  Standard  cells  should  never  be  tipped  up  for 
purposes  of  inspection  or  otherwise,  and  should  not  be  used  as  a 
source  of  current,  but  merely  for  balancing  potentials;  and  even 
here,  a  large  series  resistance  should  at  first  be  included  and  cut 
out  as  a  balance  is  approached.  Ammeters  are  instruments  for 
measuring  the  total  current  flowing,  and  should  be  connected  in 
series  with  the  circuit,  analogous  to  a  water  meter.  They  are 
most  frequently  injured  by  the  passage  of  too  large  currents. 
If  the  arrangement  of  the  apparatus  is  not  such  that  the  current 
can  approximately  be  calculated  before  the  circuit  is  closed,  a 
sufficiently  large  rheostat  should  be  included  and  cautiously  cut 
out,  the  instrument  being  watched  in  the  meantime.  Voltmeters 
are  electrical  pressure  gauges,  indicating  the  difference  of  poten- 
tial between  the  points  to  which  their  terminals  are  attached,  and 
are  accordingly  connected  in  parallel  with  the  circuit.  Most 
voltmeters  are  provided  with  two  scales;  and  in  such  cases,  one 
should  use  the  larger  first,  transferring  to  the  smaller  one  if  the 
voltage  is  found  to  be  less  than  the  lower  full  scale  reading. 
Before  leaving  the  laboratory,  return  all  apparatus  to  its  proper 
place  in  the  cases.  Wires  less  than  a  foot  long  should  be  thrown 
in  the  waste  box,  and  the  others  returned  to  their  hooks  in  the 
wire  cabinet.  Switch  board  connectors  should  be  pulled  and 
returned  to  the  proper  hooks.  Leave  the  laboratory  as  tidy  as 
you  found  it. 

7.  Notebooks. — All  data,  as  they  are  taken  during  an  experi- 
ment, should  be  recorded  in  tabular  form  in  a  rough  note- 
book with  bound  leaves.  Ascertain  from  the  instructor  specific 
directions  regarding  the  form  in  which  the  final  report  is  to  be 
made,  and  in  its  preparation  observe  the  following  outline : 


ELECTRICAL  UNITS  7 

1.  Give  name  of  experiment  and  references. 

2.  Enumerate  apparatus  used,  giving  number  of  each  piece. 

3.  Make  a  sketch  (not  a  picture)  including  all  instruments, 
resistance  boxes,  switches,  etc.,  which  will  show  the  actual  path 
of  the  current.     (Use  a  ruler  and  dividers.) 

4.  Give  the  theory  of  the  ^experiment  as  fully  as  possible, 
deriving  all  formulae  used.  ^ 

5.  Outline  briefly  thelnethod  of  procedure,  mentioning  special 
precautions  to  be  taken  and  difficulties  to  be  overcome. 

6.  Tabulate  your  data,  arranging  it  in  compact  form.     State 
the  units  in  which  your  results  are  expressed. 

7.  Plot   curves   showing   your   results   graphically,    using   as 
ordinates  the  dependent  variable.     Choose  scales  such  that  the 
curves  will  cover  as  nearly  as  possible  the  entire  sheet,  labeling 
axes  and  putting  the  scale  along  each.     Draw  in  a  smooth  curve, 
striking  an  average  between  outstanding  points. 

8.  Give  a  brief  discussion  of  results,  including  estimates  of 
accuracy  and  sources  of  error. 

9.  Answer  all  questions  asked  under  special  directions  at  the 
end  of  each  experiment. 

ELECTRICAL  AND   MAGNETIC  UNITS 

8.  Systems  of  Units.1 — There  are  two  distinct  systems  of  units 
used  in  the  measurement  of  electrical  quantities;  the  electro- 
static and  the  electromagnetic.  In  the  former,  the  fundamental 
unit  is  determined  by  means  of  the  repulsion  between  two  similar 
charges  of  electricity,  while  in  the  latter,  it  is  based  upon  the 
repulsion  of  two  similar  magnetic  poles.  Both  of  these  systems 
may  properly  be  termed  " absolute"  since  all  the  quantities 
involved  are  directly  expressible  in  terms  of  the  fundamental 
units  of  length,  mass,  and  time.  The  ratio  between  correspond- 
ing units  of  these  two  systems  is  some  power  of  the  velocity  of 
light.  In  actual  practice,  however,  neither  of  these  systems  is 
used,  since,  in  general,  the  quantities  therein  defined  are  not  of 
such  magnitudes  as  to  be  convenient  working  units.  A  third 
system,  known  as  the  ''practical  system,"  has  accordingly  been 
devised,  in  which  all  the  units  are  decimal  multiples  of  the  cor- 
responding electromagnetic  units.  The  units  of  this  system  are 

1  EVERETT,  The  C.  G.  S.  System  of  Units,  chap.  X,  XI. 
Electrical  Meterman's  Handbook,  chap.  II. 


8  ELECTRICITY  AND  MAGNETISM 

the  only  ones  to  which  names  have  been  given,  and  it  has  been 
the  custom  of  the  international  conferences  by  which  they  have 
been  defined,  to  honor  scientists,  famous  in  the  fields  in  which  the 
units  lie,  by  giving  to  them  their  names.  Electrical  quantities, 
expressed  in  the  electrostatic  and  electromagnetic  systems,  are 
designated  by  the  letters  E.S.U.  and  E.M.U.,  respectively. 

FUNDAMENTAL  ELECTRICAL  UNITS 

9.  Magnetic  Units.     Magnetic  Pole  Strength. — The  unit  mag- 
netic pole  is  a  pole  of  such  strength  that  it  repels  a  like  pole  at  a 
distance  of  one  centimeter,  in  air,  with  a  force  of  one  dyne. 

Magnetic  Field  Strength. — A  magnetic  field  of  unit  intensity  is 
a  field  that  acts  upon  a  unit  magnetic  pole  placed  in  it,  with  a 
force  of  one  dyne. 

10.  Electrostatic  Units.     Quantity. — The  electrostatic  unit  of 
quantity  is  of  such  a  magnitude  that  it  repels  a  like  quantity  at  a 
distance  of  one  centimeter,  in  air,  with  a  force  of  one  dyne. 

Current. — The  electrostatic  unit  of  current  exists  when  an 
electrostatic  unit  of  quantity  flows  past  any  plane  in  a  conductor 
per  second. 

Potential  Difference. — Unit  electrostatic  difference  of  potential 
exists  between  two  points  when  the  amount  of  work  required  to 
carry  an  electrostatic  unit  of  quantity  from  one  to  the  other  is  one 
erg. 

Resistance. — A  conductor  possesses  the  electrostatic  unit  of 
resistance  if,  when  carrying  the  electrostatic  unit  of  current,  the 
difference  of  potential  across  its  terminals  is  one  electrostatic  unit. 

Capacitance. — A  condenser  possesses  an  electrostatic  unit  of 
capacitance  if  the  electrostatic  unit  of  potential  difference 
across  its  terminals  gives  to  it  the  electrostatic  unit  of  charge. 

Inductance. — A  coil  possesses  an  electrostatic  unit  of  induc- 
tance if,  when  the  inducing  current  is  changing  at  the  rate  of  one 
electrostatic  unit  per  second,  the  induced  electromotive  force  is 
one  electrostatic  unit.  This  applies  both  to  self  and  mutual 
induction. 

11.  Electromagnetic    Units.     Current. — The    electromagnetic 
unit  of  current  is  a  current  such  that,  when  flowing  through  an 
arc  of  one  centimeter  length  of  a  circle  of  one  centimeter  radius, 
it  produces,  at  the  center,  a  unit  magnetic  field. 

Quantity. — The  electromagnetic  unit  of  quantity  is  that  quan- 


ELECTRICAL  UNITS  9 

tity  which  passes,  per  second,  any  plane  of  a  conductor  in  which 
the  electromagnetic  unit  of  current  is  flowing. 

Potential  Difference. — The  electromagnetic  unit  of  potential 
difference  exists  between  two  points  when  the  amount  of  work 
required  to  carry  the  electromagnetic  unit  of  quantity  from  one 
to  the  other  is  one  erg. 

Resistance. — A  conductor  possesses  the  electromagnetic  unit  of 
resistance,  if,  when  carrying  the'tlectromagnetic  unit  of  current, 
the  difference  of  potential  across  its  terminals  is  one  electro- 
magnetic unit. 

Capacitance. — A  condenser  possesses  the  electromagnetic  unit 
of  capacitance  if  the  electromagnetic  unit  of  potential  difference 
across  its  terminals  gives  to  it  one  electromagnetic  unit  of  charge. 

Inductance. — A  coil  possesses  an  electromagnetic  unit  of 
inductance  if,  when  the  inducing  current  varies  at  the  rate  of  one 
electromagnetic  unit  per  second,  the  induced  electromotive 
force  is  one  electromagnetic  unit. 

12.  Practical  Units.  Current. — An  ampere  is  one-tenth  of  an 
electromagnetic  unit  of  current. 

Quantity. — The  coulomb  is  the  quantity  of  electricity  which 
passes  per  second  any  plane  of  a  conductor  in  which  the  current 
is  one  ampere. 

Potential  Difference. — The  difference  of  potential  between  two 
points  is  one  volt  when  the  amount  of  work  required  to  carry  one 
coulomb  from  one  to  the  other  is  one  joule. 

Resistance. — A  conductor  possesses  a  resistance  of  one  ohm  if, 
when  carrying  a  current  of  one  ampere,  the  difference  of  poten- 
tial across  its  terminals  is  one  volt. 

Capacitance. — A  condenser  possesses  a  capacitance  of  one 
farad  if  a  difference  of  potential  of  one  volt  across  its  terminals 
gives  it  a  charge  of  one  coulomb. 

Inductance. — Two  coils  possess  a  mutual  inductance  of  one 
henry  if,  when  the  primary  current  is  changing  at  the  rate  of  one 
ampere  per  second,  the  electromotive  force  induced  in  the  second- 
ary is  one  volt. 

A  coil  possesses  one  henry  of  self-inductance  if,  when  the 
current  through  it  is  varying  at  the  rate  of  one  ampere  per  second, 
the  induced  counter  electromotive  force  is  one  volt.  One  milli- 
henry equals  0.001  henry. 

Magnetic  Flux. — The  total  flux  in  a  magnetic  circuit  is  one 
maxwell  when  it  possesses  one  magnetic  line  of  induction. 


10  ELECTRICITY  AND  MAGNETISM 

Magnetic  Induction. — The  induction  in  a  magnetic  circuit  is 
one  gauss  when  the  flux  density  is  one  maxwell  per  square 
centimeter. 

Magnetomotive  Force. — The  magnetomotive  force  of  a  magnetic 
circuit  is  one  gilbert  if  the  work  required  to  carry  a  unit  magnetic 
pole  once  around  the  circuit  is  one  erg. 

Field  Strength. — A  magnetic  field  possesses  unit  strength  if  the 
magnetomotive  force  is  one  gilbert  per  centimeter.  (This 
definition  is  identical  with  that  previously  given  for  field 
strength.) 

Reluctance. — A  magnetic  circuit  possesses  a  reluctance  of  one 
oersted  if  a  magnetomotive  force  of  one  gilbert  produces  a  flux  of 
one  maxwell. 

13.  Legal  Definitions  of  the  Practical  Units. — At  the  last 
International  Conference  on  Electrical  Units  and  Standards, 
which  met  in  London,  in  1908,  the  following  resolutions  were 
adopted,  which  have  served  as  the  basis  for  legislation  in  the 
different  countries  of  the  world  for  fixing  the  legal  definitions 
of  the  fundamental  electrical  units  now  in  force.  The  full 
report  of  this  Conference,  in  which  21  different  nations  were 
represented,  may  be  found  in  The  Electrical  Review,  vol.  63, 
(1908),  page  738. 

RESOLUTIONS 

I.  The  Conference  agrees  that  as  heretofore  the  magnitude  of 
the  fundamental  electric  units  shall  be  determined  on  the  elec- 
tromagnetic system  of  measurements  with  reference  to  the  centi- 
meter as  the  unit  of  length,  the  gram  as  the  unit  of  mass,  and  the 
second  as  the  unit  of  time. 

These  fundamental  units  are  (1)  the  ohm,  the  unit  of  electric 
resistance  which  has  the  value  of  1,000,000,000  in  terms  of  the 
centimeter  and  second ;  (2)  the  ampere,  the  unit  of  electric  current 
which  has  the  value  of  one-tenth  (0.1)  in  terms  of  the  centimeter, 
gram,  and  second;  (3)  the  volt,  the  unit  of  electromotive  force 
which  has  the  value  of  100,000,000  in  terms  of  the  centimeter, 
the  gram,  and  the  second;  (4)  the  watt,  the  unit  of  power,  which 
has  the  value  of  10,000,000  in  terms  of  the  centimeter,  the  gram, 
and  the  second. 

II.  As  a  system  of  units  representing  the  above  and  sufficiently 
near  to  them  to  be  adopted  for  the  purpose  of  electrical  measure- 
ments and  as  a  basis  for  legislation,  the  Conference  recommends 


ELECTRICAL  UNITS  11 

the  adoption  of  the  International  ohm,  the  International  ampere, 
and  the  International  volt,  defined  according  to  the  following 
definitions. 

III.  The  ohm  is  the  first  primary  unit. 

IV.  The  International  ohm  is  defined  as  the  resistance  of  a 
specified  column  of  mercury.    » 

V.  The   International   ohm  is  the   resistance   offered  to   an 
unvarying  electric  current  by  a  column  of  mercury  at  the  tem- 
perature of  melting  ice,  14.4521  grams  in  mass,  of  a  constant  cross- 
sectional  area  and  of  a  length  of  106.300  cm. 

To  determine  the  resistance  of  a  column  of  mercury  in  terms 
of  the  International  ohm,  the  procedure  to  be  followed  shall  be 
that  set  out  in  specification  I,  attached  to  these  resolutions. 

VI.  The  ampere  is  the  second  primary  unit. 

VII.  The  International  ampere  is  the  unvarying  electric  cur- 
rent which,  when  passed  through  a  solution  of  nitrate  of  silver 
in  water,  in  accordance  with  the  specification  II,  attached  to  these 
resolutions,  deposits  silver  at  the  rate  of  0.00111800  of  a  gram  per 
second. 

VIII.  The  International  volt  is  the  electrical  pressure  which, 
when  steadily  applied  to  a  conductor  whose  resistance  is  one 
International  ohm,  will  produce  a  current  of  one  International 
ampere. 

IX.  The  International  watt  is  the  energy  expended  per  second 
by  an  unvarying  electric  current  of  one  International  ampere 
under  an  electric  pressure  of  one  International  volt. 

The  Conference  recommends  the  use  of  the  Weston  Normal 
Cell  as  a  convenint  method  of  measuring  both  electromotive 
force  and  current,  and  when  set  up  under  the  conditions  specified 
in  schedule  C,  may  be  taken,  provisionally,  as  having,  at  a  tem- 
perature of  20°  C.,  an  E.M.F.  of  1.0184  volts. 

14.  The  New  Value  of  the  Weston  Standard  Cell.— Since  the 
meeting  of  the  London  Conference,  a  large  amount  of  research 
has  been  carried  on  at  the  Bureau  of  Standards  at  Washington 
on  the  Weston  Cell  and  the  electrochemical  equivalent  of  sil- 
ver; and  it  has  been  found  that  the  electromotive  force  of  this 
cell,  in  terms  of  the  International  ohm  and  International  ampere, 
is,  within  one  part  in  10,000, 

E  =  1.0183  International  volts  at  20°  C., 
and  this  value  was  adopted  by  the  Bureau  of  Standards  Jan.  1, 


12 


ELECTRICITY  AND  MAGNETISM 


1911.     The  formula  for  the  temperature  coefficient  of  the  Weston 
Cell  adopted  by  the  London  Conference,  based  on  the  investiga- 
tions of  the  Bureau  of  Standards,  is  as  follows : 
Et  =  #20  -  .0000406  (t  -  20°)  -  .00000095  (t  -  20)2 

+  .00000001  (t  -  20) 3     (1) 

15.  Ratios  of  the  Electrical  Units. — For  convenience  of  com- 
parison the  dimensions  of  the  electrostatic  and  electromagnetic 
units  are  given  below.  The  dimensions  of  the  dielectric  constant 
and  the  permeability  are  unknown  and  are  inserted  in  the  formula 
as  K  and  n,  respectively.  All  that  is  known  concerning  the 

nature  of  these  quantities  is  that     / —   equals  v,   equals   3   X 

1010  cm.  per  second,  the  velocity  of  light  in  free  space.  The 
last  column  gives  the  ratio  of  the  corresponding  units  in  the  two 
systems,  in  terms  of  v. 


Unit 


Electro- 
magnetic 


Electro- 
static 


Electro- 
magnetic 


Electro- 
static 


E.M.U. 


E.S.U. 


Quantity . . . 

Current 

Pot.  diff. . . . 
Resistance . 
Capacity. . . 
Inductance . 


I 


[I*] 


The  following  table  gives  the  practical  units  in  terms  of  the  corre- 
sponding units  of  both  the  electromagnetic  and  the  electrostatic  systems: 


1  Ampere 
1  Coulomb 

1  Volt 

1  Ohm 
1  Farad 


=  10-1  E.M.U.'s  =  3  X  109  E.S.U.'s 
=  10-1  E.M.U.'s  =  3  X  109  E.S.U.'s 
1 


108     E.M.U.'s 
10*     E.M.U.'s 


3  X  102 

1 


E.S.U.'s 
E.S.U.'s 


9  X  1011 

=  10-9  E.M.U.'s  =  9  X  10"  E.S.U.'s 
9  X  105  E.S.U.'s 
1 


1  Microfarad  =  KT15  E.M.U.'s 
1  Henry  =  109     E.M.U.'s 


9  X  10" 


E.S.U.'s 


ELECTRICAL  UNITS  13 


THE  RATIONALIZED  PRACTICAL  SYSTEM  OF  UNITS 

16.  Advantages  of  the  Rationalized  System. — In  the  discussion 
of  the  practical  system  it  was  pointed  out  that  our  present  work- 
ing units  are  decimal  multiples  of  the  corresponding  units  of  the 
electromagnetic  system.     In  (fixing  these  ratios  the  international 
conferences  have  selected  valuesdn  such  a  way  that  the  electrical 
quantities  commonly  measured  are  expressed  by  numbers  of 
ordinary  magnitude.     This,  in  reality,  constitutes  a  mixed  system 
of  units,  and,  as  a  result,  many  of  the  formula?  used  in  every  day 
calculations  contain  factors  such  as  10"1,  108,  109,  etc.     Again,  a 
system  based  upon  the  unit  magnetic  pole  and  the  unit  electric 
charge  as  given  in  paragraphs  9  and  10,  respectively,  inevitably 
leads  to  many  formulae  in  which  the  factor  4w  appears. 

It  has  been  pointed  out  by  Perry1  and  by  Fessenden2  that  by 
properly  choosing  new  units  for  magnetomotive  force  and  field 
strength ,  and  by  submerging  the  factor  4?r  in  the  arbitrary  constants 
defining  the  dielectric  and  magnetic  properties  of  materials,  that 
these  objectionable  factors  may  be  eliminated,  and  all  that 
Heaviside  sought  to  accomplish  by  his  "  Rationalized  System 
of  Units,"  realized.  In  an  admirable  paper  entitled  "  A  Digest  of 
the  Relations  between  the  Electrical  Units  and  the  Laws  under- 
lying the  Units,"  Bennett3  has  carried  out  the  suggestions  of 
Perry  and  Fessenden  and  has  developed  a  consistent  series  of 
defining  equations  and  working  formulae  in  which  the  objection- 
able factors  are  suppressed,  and  has  clearly  set  forth  the  relations 
between  the  units  of  the  different  systems.  In  this  treatment,  a 
new  unit  of  force,  the  " Dyne-seven"  (equal  to  107  dynes)  has 
been  introduced.  The  advantage  of  this  unit  is  obvious,  since, 
when  acting  through  one  centimeter,  it  performs  one  joule  of 
work. 

17.  Definitions.     1.   Unit  Quantity  of  Electricity. — The  method 
followed  here  is  similar  to  that  of  the  electrostatic  system  in 
that  the  unit  of  charge  is  taken  as  the  fundamental  unit,  and  its 
magnitude  is  arrived  at  by  an  application  of  Coulomb's  law, 

namely, 

n  r\ 
F  = 


kd* 

1  PERRY,  Electrician,  vol.  27,  1891,  p.  355. 

2  FESSENDEN,  Electrical  World,  vol.  34,  1899,  p.  901. 

3  Bull.  880,  Univ.  of  Wisconsin. 


14  ELECTRICITY  AND  MAGNETISM 

Where  Qi  and  Q2  are  two  charges  of  electricity  placed  d  cm.  apart. 
The  quantity  fc  is  a  constant  depending  upon  the  medium  in 
which  the  charges  are  placed,  and  for  free  space  is  arbitrarily 

put  equal  to  Q  v  lnil-     If  Qi  =  Q2  =  1  and  d  =  1,  then  F  = 
y  /\  JLU 

9  X  1011  dyne  sevens.  Accordingly  the  coulomb  is  that  quan- 
tity of  electricity  which  repels  a  similar  quantity  at  a  distance  of 
one  centimeter  in  a  vacuum  with  a  force  of  9  X  1011  dyne  sevens. 
The  coulomb,  as  thus  defined,  is  identical  with  that  defined  in  the 
practical  system  of  Art.  12. 

2.  Permittivity  of  a  Medium. — The  factor  k  in  the  expression 
for  Coulomb's  law  is  called  the  dielectric  constant  of  the  medium. 

k 
Since  in  many  formulae  the  factor  j-  appears,  it  is  expedient  to 

replace  k  by  a  new  medium  constant  defined  by  the  relation 

A 

and  Coulomb's  law  is  then  written 

n.n. 
F  = 


4-rrpdz 

The  quantity  p  is  called  the  permittivity  of  the  medium,  and  for 
free  space  has  the  numerical  value 

*  -  I  -  idhon  =  8M  x  10-u 

The  relative  permittivity  of  a  substance  is  the  ratio  of  the  per- 
mittivity of  the  substance  to  the  permittivity  of  free  space,  and  is 
thus  numerically  equal  to  the  dielectric  constant  or  specific 
inductance  capacity  as  ordinarily  defined. 

3.  Unit  of  Current;  the  Ampere. — A  current  of  one  ampere  is 
flowing  in  a  circuit  if  the  quantity  passing  any  plane  in  the  circuit 
per  second  is  one  coulomb. 

4.  Unit  of  Potential  Difference;  the  Volt. — A  difference  of  poten- 
tial of  one  volt  exists  between  two  points  if  the  work  required  to 
carry  one  coulomb  from  one  point  to  the  other  is  one  joule. 

5.  Unit  of  Resistance;  the  Ohm. — A  conductor  has  a  resistance 
of  one  ohm  if  a  difference  of  potential  between  its  terminals  of 
one  volt  maintains  a  current  of  one  ampere. 

6.  Unit  of  Capacitance;  the  Farad. — A  condenser  has  a  capaci- 
tance of  one  farad  if  a  charge  of  one  coulomb  produces  differences 
of  potential  between  its  plates  of  one  volt. 


ELECTRICAL  UNITS  15 

7.  Unit  of  Inductance;  the  Henry. — A  coil  has  an  inductance  of 
one  henry  if  a  current  through  it,  changing  at  the  rate  of  one 
ampere  per  second,  induces  within  it  an  E.M.F.  of  one  volt. 

8.  Line  of  Magnetic  Intensity. — By  a  line  of  magnetic  intensity 
or  a  line  of  force  in  a  magnetic  field  is  meant  any  line  which  is 
traced  out  by  the  center  point  .-of  a  small  plane  direction -finding 
coil,1  as  the  coil  is  moved  in  the  direction  pointed  out  by  its 
normal  axis.     Such  lines  are  always  found  to  be  closed  loops, 
which    either    link    with    electric    currents    or    pass    through 
magnets. 

9.  Magnetic  Flux  Density. — The  magnetic  flux  density,  B,  at 
a  point  in  a  magnetic  field  is  defined  as  a  vector  quantity  whose 
direction  is  the  positive  direction  along  the  line  of  magnetic 
intensity  passing  through  the  point,  and  whose  magnitude  is  equal 
to  the  force  upon  a  straight  wire  one  centimeter  in  length  carrying 
a  current  of  one  ampere,  the  direction  of  the  wire  making  a  right 
angle  with  a  line  of  magnetic  intensity  through  the  point. 

Unit  of  Flux  Density. — The  Weber  per  square  centimeter. — If  a 
wire  one  centimeter  in  length  carrying  a  current  of  one  ampere , 
in  a  direction  at  right  angles  to  the  lines  of  magnetic  intensity  is 
acted  upon  by  a  force  of  one  dyne  seven,  the  flux  density  is  one 
weber  per  square  centimeter.  One  weber  per  square  centimeter 
equals  108  gauss. 

10.  Relation  Between  the  Magnetic  Flux  Density  and  the  Current 
Causing  the  Field. — Experimental  measurements  show  that  at 
any  point  in  a  field,  free  from  iron,  the  value  of  the  magnetic  flux 
density,  J5,  is  directly  proportional  to  the  value  of  the  current 
producing  the  field.     For  the  special  case  of  an  annular  ring 
uniformly  wound  with  a  coil  of  N  turns,  carrying  a  current  7, 
experimental  measurements  show  that  the  lines  of  magnetic  in  ten- 
sity are  circles  lying  within  the  ring  as  illustrated  in  Fig.  57  and 
that  the  value  of  the  flux  density,  B,  is  uniform  along  each  circle 
and  has  the  value 

B  =     N- 
L 

1  A  direction-finding  coil  is  a  small  plane  circular  coil  carrying  a  continuous 
current.  The  coil  is  so  mounted  on  gimbals  that  its  normal  axis  is  free  to 
take  any  direction.  The  normal  axis  is  a  line  perpendicular  to  the  plane  of 
the  coil  at  its  center.  The  positive  direction  along  the  normal  axis  is  defined 
to  bear  the  same  relation  to  the  direction  of  the  current  around  the  coil  that 
the  direction  of  advance  of  a  right-hand  screw  bears  to  its  direction  of  rota- 
tion. This  is  called  the  right-hand  screw  convention. 


16  ELECTRICITY  AND  MAGNETISM 

in  which  L  is  the  length  of  the  circle.  ^  is  a  constant  having  the 
value  1.257  X  10~8  for  all  except  ferromagnetic  materials. 

11.  Permeability. — The  constant  /*,  which  appears  in  the  equa- 
tion expressing  the  relation  between  the  flux  density  and  the 
current,  is  called  the  permeability  of  the  medium  in  which  the 
magnetic  field  is  set  up.     It  is  a  constant  analagous  to  conduc- 
tivity in  the  conducting  field  and  to  permittivity  in  the  electric 
field.     This  unit  is  called  the  weber  per  ampere  turn  per  centi- 
meter and  is  equal  to  4ir  X  10~9  units  of  permeability  as  defined 
in  the  unrationalized  practical  system. 

12.  Magnetic  Intensity. — The  defining  equation  of  (10)  may 
be  written  in  the  form 

B      NI 
M   =     L 

D 

The  expression       appears  in  so  many  calculations  dealing  with 

magnetic  fields  that,  for  the  sake  of  convenience,  the  name  "  mag- 
netic intensity"  or  "strength  of  field"  is  given  to  it.  It  is  seen 
to  be  equal  to  the  number  of  ampere  turns  per  centimeter.  This 
unit  of  field  strength  is  called  the  ampere  turn  per  centimeter  and 

is  equal  to  T-  gilberts  per  centimeter. 

13.  Magneto-motive  Force. — The  line  integral  J*Hdl  for  any 
closed  magnetic  circuit  is  called  the  magneto-motive  force  for 
that  circuit.     For  the  simple  circuit  of  Fig.  57  we  have  fHdl  = 
HL  =  NI.     The  unit  of  magneto-motive  force  is  the  "Ampere 

4rr 
Turn"  and  is  equal  to  j~  gilberts. 

14.  Reluctance,  Ampere  Turn  per  Weber. — A  magnetic  circuit 
possesses  a  reluctance  of  one  ampere  turn  per  weber  if  a  magneto- 
motive force  of  one  ampere  turn  produces  a  flux  of  one  weber. 

109 
One  ampere  turn  per  weber  equals  -.-    oersteds. 


CHAPTER  II 
GALVANOMETERS1 

18.  Description  of  a  Galvanometer. — A  galvanometer  is  an 
instrument  for  the  detection  and  measurement  of  very  small 
electric  currents.     Strictly  speaking,  when  used  merely  for  the 
detection  of  an  electric  current,  as,  for  example,  in  determining 
the  balance  condition  for  a  Wheatstone  bridge  or  a      A  m 
potentiometer,    it   should   be  called  a  galvanoscope, 

and  the  term  galvanometer  restricted  to  the  case  in 
which  it  is  standardized  and  used  for  the  accurate 
measurement  of  currents.  The  fundamental  principle 
upon  which  all  galvanometers  operate  is  the  reaction 
between  a  current  and  a  magnetic  field,  one  of  which 
is  fixed  and  the  other  movable.  There  are  two  types  j 
of  instruments,  named  after  their  originators,  and 
known  respectively  as  the  Thomson  and  the 
D'Arsonval  types. 

19.  Thomson    Galvanometer. — The  Thomson  gal- 
vanometer was  invented  by  William  Thomson  (Lord 
Kelvin)  and  was  first  used  as  a  detecting  instrument     , 
in  connection  with  the  trans-Atlantic  cable.     It  uses 

fixed  coils  and  moving  magnets,  the  axes  of  which       FlG  5 
are  placed  at  right  angles  to  the  fields  produced  by       Astatic 
the  current  in  the  coils.     A  high  sensitivity  requires,       system, 
among  other  things,  that  the  restoring  torque  on  the 
moving  system  should  be  as  small  as  possible.     This  is  accom- 
plished by  use  of  the  so-called  astatic  system  which  is  illustrated 
in  Fig.    5.     A   rigid  rod,  BC,  usually  a  slender  glass  tube,  is 
suspended  by  a  very  fine  quartz  fibre  A  B.     This  rod  carries 
two  systems  of  magnets  NS  placed  with  their  planes  accurately 
parallel,  but  with  polarities  reversed.     If  the  magnetic  moments 
of  the  two  groups  of  magnets  are  equal,  then  when  the  system 

1  LAWS,  Electrical  Measurements,  chap.  I. 
BROOKS  and  POYSER,  Magnetism  and  Electricity,  chap.  XIX. 
HADLEY,  Magnetism  and  Electricity,  chap.  XVI. 
2  17 


18  ELECTRICITY  AND  MAGNETISM 

is  placed  in  a  uniform  magnetic  field,  it  will  remain  in  any 
position  in  which  it  is  placed,  since  the  torque  on  one  group  of 
magnets  is  balanced  by  that  on  the  other. 

The  fixed  coils  which  carry  the  current  to  be  measured  are 
wound  in  opposite  directions  so  that  the  reactions  of  their  fields 
upon  the  magnets  of  the  moving  system  give  torques  in  the  same 
direction.  By  making  the  system  very  light,  e.g.,  a  few  milli- 
grams, and  by  using  a  very  fine  quartz  fibre  for  suspension,  it  is 
possible,  with  this  type  of  instrument,  to  measure  currents  of  the 
order  of  10~12  amperes.  Since  the  fields  due  to  currents  of  such 
magnitudes  are  very  weak,  slight  gradients  in  the  external  field 
produce  relatively  large  differences  in  the  torques  upon  the 
upper  and  lower  magnet  systems,  and  unsteadiness  of  the  zero 
position  results.  Galvanometers  of  this  type  must,  therefore,  be 
carefully  shielded  magnetically. 

Magnetic  shields1  may  be  either  spherical  or  cylindrical  in 
shape,  but  since  no  openings  may  be  permitted  without  serious 
reduction  in  effectiveness,  the  latter  form  is  usually  employed. 
It  has  been  found  that  if  the  iron  is  all  concentrated  in  a  single 
cylindrical  shell  having  an  outside  diameter  five  times  that  of  the 
inner,  the  effectiveness  is  98  per  cent  of  that  of  a  shield  having 
an  infinite  thickness.  Furthermore,  for  a  given  amount  of  iron, 
the  effectiveness  is  greatly  increased  by  using  several  concentric 
cylinders.  For  extreme  sensitivity,  Thomson  galvanometers  are 
made  very  small,  and  the  coils  are  often  mounted  in  a  solid  iron 
container  made  by  splitting  a  soft  iron  rod  longitudinally  and 
drilling  small  holes  in  each  half  to  receive  the  coils. 

20.  D'Arsonval  Galvanometer. — The  D' Arson val  galvano- 
meter consists  of  a  fixed,  permanent  horse-shoe  magnet  and  a 
light  coil  suspended  between  the  pole  pieces  by  a  fine  phosphor- 
bronze  ribbon,  the  plane  of  the  coil  being  parallel  to  the  direction 
of  the  field.  The  current  is  led  to  the  coil  by  the  supporting 
ribbon  and  away  by  a  helix  of  the  same  material  attached  at  the 
bottom.  While  this  type  of  instrument  cannot  be  made  as 
sensitive  as  the  Thomson,  it  has  the  following  special  advantages : 
(a)  The  deflections  are  but  little  affected  by  variations  in  the 
external  magnetic  field;  (b)  the  instrument  may  face  in  any  direc- 
tion; (c)  the  moving  system  may  be  made  aperiodic,  thus  avoiding 
loss  of  time  in  waiting  for  it  to  come  to  rest.  For  these  reasons, 
except  where  extreme  sensitivity  is  required,  the  D'Arsonval 

1  WILLS,  Physical  Review,  vol.  24,  1907,  p.  243. 


GALVANOMETERS 


19 


galvanometer  has  practically  replaced  the  Thomson  for  general 
laboratory  work. 

There  are  two  distinct  purposes  for  which  galvanometers  are 
used:  (a)  The  measurement  of  small  currents,  and  (6)  the 
measurement  of  small  quantities  of  electricity,  such  as  are 
obtained  by  the  discharge  of  condensers.  When  designed  for  the 
first  purpose,  they  are  called  ".current  galvanometers"  and  for 
the  second,  " ballistic  galvanometers." 


FIG.  6. — High  sensitivity  galvanometer  with  cover  removed. 

21.  The  Current  Galvanometer. — Figure  6  shows  a  high 
sensitivity  current  galvanometer  manufactured  by  the  Leeds  and 
Northrup  Company.  The  permanent  magnet  is  mounted  in  a 
vertical  position  and  is  provided  with  pole  tips  shaped  so  as  to 
give  a  nearly  cylindrical  gap  between  them.  Coaxial  with  this 
gap  is  placed  a  cylinder  of  soft  iron  and  the  coil  rotates  in  the 
annular  space  thus  formed.  The  suspension  is  carried  on  a  rod 
supported  by  a  bracket  from  the  magnet.  A  set  screw  permits 
a  vertical  adjustment  of  the  coil  and  the  knurled  head,  which 
projects  through  the  top  of  the  case,  gives  a  rough  adjustment  for 
zero  position  on  the  scale.  A  slow  motion  screw  at  the  base  of 
the  instrument  gives  the  final  zero  setting  The  axis  of  the  coil 


20  ELECTRICITY  AND  MAGNETISM 

is  made  to  coincide  with  that  of  the  gap  by  means  of  three  level- 
ing screws  which  support  the  instrument.  These  screws  are 
turned  by  heavy  vulcanite  nuts  which  give,  at  the  same  time, 
good  insulation  from  ground.  The  right  hand  screw  at  the  top 
operates  an  arresting  device  by  means  of  which  the  weight  of  the 
coil  may  be  taken  off  the  suspension  when  the  instrument  is 
being  moved.  A  cylindrical  case,  provided  with  a  window  to 
pass  light  to  and  from  the  mirror,  protects  the  system  against  air 
currents. 

22.  Galvanometer  Sensitivity. — If  several  galvanometers, 
selected  at  random,  are  connected  in  series  and  a  definite  current 
is  sent  through  them,  it  will  be  found  that  there  are  marked 
differences  in  the  response  made  by  the  individual  instruments. 
Those  showing  greater  responses  are  said  to  have  higher  sensi- 
tivities. The  indication  of  a  galvanometer  is  usually  read  by 
means  of  a  beam  of  light  reflected  from  a  mirror,  attached  to  the 
moving  system,  on  a  fixed  scale.  Obviously,  for  a  given  motion 
of  the  system,  the  indication  will  be  proportional  to  the  distance 
from  mirror  to  scale,  and  so  it  is  customary,  when  comparing 
galvanometers,  to  place  the  scale  at  a  distance  of  one  meter, 
and  to  read  the  deflection  in  millimeters.  The  sensitivity  of 
galvanometers  is  defined  in  a  number  of  ways  among  which  the 
following  are  the  most  common : 

(a)  Microampere  Sensitivity. — This  is  defined  as  the  deflection 
in  millimeters  of  a  spot  of  light  on  a  scale  one  meter  from  the 
mirror  when  the  deflecting  current  is  one  microampere. 

(b)  Microvolt  Sensitivity. — By  this  is  meant  the  deflection  in 
millimeters  of  a  spot  of  light  on  a  scale  one  meter  from  the  mirror 
when   an   E.M.F.    of   one   microvolt   is   impressed   across   the 
terminals  of  the  galvanometer. 

(c)  Megohm  Sensitivity. — By  this  is  understood  the  number  of 
megohms  which  must  be  placed  in  the  galvanometer  circuit  in 
order  that  with  an  impressed  E.M.F.  of  one  volt  there  results  a 
deflection  of  one  millimeter  on  the  scale  whose  distance  is  one 
meter. 

The  dependence  of  the  sensitivities,  as  just  defined,  upon  the 
constants  of  the  instrument  and  the  relations  between  them  may 
be  understood  from  the  following  considerations.  It  will  be 
assumed  that  the  coil  is  rectangular  in  shape  and  that  it  is  so 
supported  as  to  be  capable  of  rotation  about  a  vertical  axis  of 
symmetry.  It  will  also  be  assumed  that  the  field  is  radial, 


GALVANOMETERS  21 

uniform  and  horizontal,  as  shown  in  Fig.  7.  A  field  of  this  sort  is 
obtained  by  means  of  a  cylindrical  core  between  properly  shaped 
pole  pieces,  and  has  the  advantage  that,  for  a  constant  current 
through  the  coil,  the  torque  is  independent  of  its  angular  position. 

Let  I  be  the  length  of  the  coil;  b  its  width;  n  the  number  of 
turns;  and  H  the  strength  of  the  field  in  which  it  is  placed. 
Calling  T  the  torque  on  the  coil  when  the  current  flowing  through 
it  is  ij  we  have,  if  c,0,s^units  arfc  used, 

T  =  Hinlb  (1) 

The  quantity  nib,  that  is,  the  product  of  the  number  of  turns 
and  the  area  of  the  coil,  is  frequently  called  the  "  equivalent 
winding  surface."  Designating  this  by  E  we  have 

T  =  HiE  =  Ci  (2) 

where  C,  equal  to  HE,  is  the  torque  for  unit  current  and  is  called 
the  " Dynamic  Constant"  for  the  instrument. 


FIG.  7. — Diagram  of  moving  coil  galvanometer. 

As  the  coil  rotates,  it  twists  the  supporting  metallic  ribbon 
which  exerts  an  elastic  counter  torque  proportional  to  the  angle 
of  twist,  and  the  coil  takes  an  equilibrium  position  such  that  the 
two  torques  balance  each  other.  Designating  by  T  the  constant 
of  the  suspension,  that  is,  the  restoring  torque  when  it  is  twisted 
through  an  angle  of  one  radian,  the  angular  deflection  B  for  a 
given  current  i  satisfies  the  relation 

rO  =  Ci  (3) 

Letting  A  equal  the  angular  displacement  resulting  from  unit 
current  we  have 

*  =  ;  =  -  w 

A  is  the  angular  displacement  in  radians  resulting  from  one 
C.G.S.  unit  of  current  and  is,  accordingly,  the  current  sensitivity 
in  C.G.S.  units.  The  microampere  sensitivity,  as  defined  above, 
may  be  obtained  from  eq.  (4)  in  the  following  manner:  For 


22  ELECTRICITY  AND  MAGNETISM 

small  deflections,  the  angular  displacement  of  the  system  is 
proportional  to  the  linear  displacement  of  the  spot  of  light  along 
the  scale.  Moreover,  the  angular  displacement  of  the  reflected 
beam  is  twice  that  of  the  reflecting  mirror.  Accordingly,  a 
deflection  A  in  radians  is  equivalent  to  a  deflection  2,00(14.  when 
expressed  as  the  deflection  in  millimeters  of  a  spot  of  light  along 
a  scale  at  a  distance  of  one  meter  from  the  mirror. 

Again,  if  the  current  is  measured  in  microamperes  instead  of 
C.G.S.  units,  it  follows,  since  1  microampere  is  10~7  C.G.S.  units, 
that  the  right-hand  member  of  eq.  (4)  must  be  divided  by  107  for 
this  case.  Therefore,  replacing  A  by  its  value  S  divided  by  2,000, 
where  S  is  the  deflection  in  millimeters  due  to  one  microampere, 
we  have 

S  =  ~  X  2  X  10~4  (5) 

This  is  the  microampere  sensitivity  and  is  seen  to  be  .0002  times 
the  ratio  of  the  dynamic  constant  to  the  suspension  constant. 
It  is  easily  seen  that  the  megohm  sensitivity  defined  above  is 
numerically  equal  to  the  microampere  sensitivity  just  discussed. 
For,  if  S  is  the  deflection  in  millimeters  due  to  one  microampere, 
then  the  current  in  amperes  required  for  a  deflection  of  milli- 

meter is  0  ir>6-     Let  M  ke  the  rnegohm  sensitivity;  that  is,  the 

number  of  megohms  placed  in  series  such  that  the  deflection  is  1 
millimeter  when  the  E.M.F.  is  1  volt.  By  Ohms  law, 

i  =  slo*  =  206*  whence  M  =  S  (6) 

To  obtain  the  relation  between  the  microampere  and  the  micro- 
volt sensitivity,  let  it  be  supposed  that  a  difference  of  potential 
of  1  microvolt  is  impressed  across  the  galvanometer.  The  cur- 
rent i  in  microamperes  is  given  by 


where  R  is  the  resistance  of  the  galvanometer.     The  resulting 
deflection  V,  or  the  microvolt  sensitivity  is,  accordingly, 

V  =  Si  =  I  (10) 

Thus  the  microvolt  sensitivity  is  obtained  by  dividing  the  micro- 
ampere sensitivity  by  the  resistance  of  the  coil. 

23.  Figure  of  Merit.  —  While  the  definitions  of  galvanometer 
sensitivity  given  above  are  convenient  for  distinguishing  the 


GALVANOMETERS 


23 


properties  of  one  galvanometer  from  another,  they  are  not  well 
suited  to  the  practical  case  in  which  the  instrument  is  to  be  stand- 
ardized and  used  for  the  measurement  of  currents.  Here  it  is 
simpler  to  use  the  relation 


If  d  is  the  deflection  in  millimeters  at  a  meter  distance  and  i 
is  in  amperes,  F  is  called  the  "figure  of  merit"  or  simply  the 
"constant"  of  the  galvanometer  and  .is  defined  as  the  current  in 
amperes  required  to  produce 
a  deflection  of  1  millimeter 
at  a  distance  of  1  meter. 
The  smaller  F,  the  greater  is 
the  sensitivity  of  the  instru- 
ment. 

To  determine  the  figure  of 
merit  of  a  galvanometer  it  is 
merely  necessary  to  pass 
known  currents  through  the 
instrument  and  measure  the 
deflections  they  produce. 
These  currents  may  be  sup- 


FIG.  8. — Connection  for  figure  of  merit. 


plied  through  a  standardized 
variable  resistance  by  a  cell 
of  known  E.M.F.,  and  computed  by  Ohm's  law.  Since,  for 
most  galvanometers,  the  required  current  is  very  small,  the 
arrangement  shown  in  Fig.  8  is  generally  employed.  By  making 
P  small,  usually  10  or  100  ohms,  and  Q  large,  1,000  or  10,000 
ohms,  only  a  small  fraction  of  the  E.M.F.  of  the  cell  is  effective 
in  sending  a  current  to  the  galvanometer  G,  and  this  current 
may  be  still  further  reduced  by  making  R  large.  If  R  +  G  is 
large  in  comparison  to  P,  the  fall  of  potential  across  P  is 

*=w^E  (12) 


where  E  =  E.M.F.  of  cell  read  by   the  voltmeter   VM. 
current  i  through  the  galvanometer  is 

.  e  P          E 

~R+G~P+QR+G 
and  the  constant  F  is  given  by 

PE  1 


F  = 


Q)(R+G)d 


The 


(13) 


(14) 


24  ELECTRICITY  AND  MAGNETISM 

Since,  in  no  galvanometer,  is  the  deflection  strictly  proportional 
to  current,  it  is  necessary,  in  making  a  standardization,  to  use 
currents  giving  deflections  over  the  entire  range  for  which  the 
instrument  is  to  be  used,  determining  from  each  a  value  of  F 
which,  when  plotted  as  ordinates  against  d,  as  abscissas,  gives  a 
working  curve  showing  F  as  a  function  of  the  deflections. 

24.  The  Ballistic  Galvanometer. — The  ballistic  galvanometer, 
which  may  be  of  either  the  moving  coil  or  the  moving  magnet 
type,  differs  from  the  current  galvanometer  in  that  its  moving 
system  has  a  large  moment  of  inertia,  giving  it  a  long  period  of 
vibration.     If,  while  the  system  is  at  rest,  a  small  quantity  of 
electricity,  such  as  a  condenser  charge,  is  suddenly  passed  through 
it,  during  the  small  interval  of  time  that  this  electricity  is  flowing, 
there  will  be  a  torque  acting  on  the  system.     This  torque  must 
be  of  very  short  duration  as  compared  with  the  time  required 
for  the  complete  swing  of  the  instrument,  and  is  called  an  impul- 
sive torque.     The  system  is  thus  given  an  angular  velocity,  and 
an  application  of  the  laws  of  mechanics  shows  that  the  amplitude 
of  the  first  ballistic  throw  is  a  measure  of  the  impulsive  torque 
applied,  and  hence  of  the  quantity  of  electricity  that  has  passed. 
The  ballistic  galvanometer  is,  then,  an  integrating  rather  than  an 
indicating  instrument.     The  rotational  energy  of  the  moving 
system  is  consumed  in  two  ways:  (a)  The  air  surrounding  the 
system  is  set  in  motion;  (b)  the  relative  motion  of  the  coil  and 
magnet  induces  a  current  in  the  coil,  if  the  circuit  is  closed.     Since 
the  system  is  thus  losing  energy,  each  succeeding  swing  is  less 
than  the  preceding  one,  the  instrument  comes  gradually  to  rest, 
and  the  motion  is  said  to  be  "  damped."     If  the  resistance  across 
the  galvanometer  terminals  is  very  large,  the  system  will  make 
several  swings  before  coining  to  rest.     If  the  resistance  is  small, 
the  system  will  not  vibrate  at  all,  but  will  come  to  rest  slowly. 
If,  however,  it  is  of  the  proper  value,  the  motion  may  be  just 
aperiodic;  that  is,  it  will  not  swing  past  zero,  but  will  return  to 
zero  in  the  shortest  time.     The  instrument  is  then  said  to  be 
" critically"  damped,  and  the  resistance  required  is  called  the 
"  critical  resistance."     In  many  instruments,  the  moving  coil  is 
wound  on  a  closed  copper  form  in  which  currents  are  induced  as 
it  swings,  thus  making  it  nearly  aperiodic  on  open  circuit. 

25.  Constant  of  a  Ballistic  Galvanometer. — A  study  of  the 
equation  of  motion  of  the  ballistic  galvanometer  shows  that,  no 
matter  what  the  damping  may  be,  whether  zero  or  so  great  that 


GALVANOMETERS  25 

the  motion  is  aperiodic,  the  first  throw  is  proportional  to  the 
quantity  of  electricity  discharged  through  it,  the  only  limitation 
being  that  this  discharge  must  take  place  before  the  system  moves 
appreciably  from  its  zero  position.  If  the  throw  is  small,  so 
that  the  tangent  is  proportional  to  the  angle,  this  fact  may  be 
expressed  thus 

Q  =  Kd  (15) 

where  Q  is  the  quantity  of  electricity,  d  the  deflection  as  read  by 
a  mirror  and  scale,  and  K  a  constant  depending  upon  the  sensi- 
tiveness of  the  instrument,  numerically  equal  to  the  quantity 
necessary  to  give  unit  deflection.  The  smaller  K,  the  greater  is 
the  sensitiveness  of  the  instrument.  If,  then,  K  is  known,  we 
have  a  means  of  measuring  small  quantities  of  electricity 

26.  Theory  of  the  Undamped  Ballistic  Galvanometer. — It 
will  be  assumed  that  the  galvanometer  is  of  the  D' Arson val  type 
and  that  the  field  in  which  the  coil  moves  is  radial  and  uniform. 
It  will  also  be  assumed  that  the  duration  of  the  discharge  is  short 
compared  to  the  time  required  for  the  first  ballistic  throw  to 
take  place.  The  conditions  under  consideration,  then,  are  these : 
A  small  quantity  of  electricity,  such  as  the  charge  of  a  condenser, 
is  passed  through  the  coil.  While  the  current  is  flowing,  the 
reaction  between  the  current  and  the  field  produces  a  torque  on 
the  coil  which  starts  it  rotating.  Although  the  duration  of  this 
torque  is  very  short,  the  coil  has,  nevertheless,  acquired  a  certain 
kinetic  energy,  and  its  motion  is  opposed  only  by  the  counter 
torque  of  the  suspension,  since  we  are  neglecting  damping.  It 
will  continue  to  rotate  until  its  energy  has  been  transferred  to 
the  suspension  where  it  is  stored  as  potential  energy  of  elastic 
deformation.  The  coil  then  starts  swinging  in  the  reverse  direc- 
tion and  when  it  passes  through  its  zero  position,  it  again  pos- 
sesses the  same  kinetic  energy  that  it  had  originally,  and  will 
continue  to  oscillate  indefinitely. 

Let  7  be  the  moment  of  inertia  of  the  coil,  o>  its  angular  velo- 
city, a  its  angular  acceleration,  and  0  its  angular  deflection  at  any 
instant.  As  in  the  discussion  of  the  current  galvanometer,  let  C 
be  the  coil  constant,  that  is,  the  torque  produced  by  unit  current, 
and  let  r  be  the  suspension  constant,  that  is,  the  counter  torque 
for  a  twist  of  one  radian.  At  any  instant  during  the  discharge, 
the  equation  of  motion  for  the  system  is 

Ci  -rd  =  la  (16) 


26  ELECTRICITY  AND  MAGNETISM 

Since  we  are  assuming  that  the  discharge  takes  place  before  the 
coil  swings  appreciably  from  its  zero  position,  the  second  term  on 
the  left  hand  side  may  be  neglected;  and,  writing  for  a  its  value, 

d*S 

-Tp>  we  have 

«  =  /g  (17) 

Let  tr  be  the  time  required  for  the  discharge  to  take  place.     Then 


Carrying  out  this  integration  and  letting  a/  be  the  angular  velo- 
city at  the  time  t',  we  have 

CQ  =  Ao'  (19) 

Where  Q  is  the  quantity  of  electricity  which  passed  through  the 
coil.     The  kinetic  energy  thus  acquired  by  the  coil  is 

Energy  =  W*  -  H  ^  (20) 


If  the  coil  swings  through  an  angle  8\,  the  potential  energy  of 
elastic  deformation  is 


fit 

=  T      ede  = 

Jo 


W  =  r  \0de  =  Kr<V  (21) 


Since  this  is  equal  to  the  initial  kinetic  energy   of  rotation, 
there  results 


whence 

Q-^r*.  ,  (22) 

Inasmuch  as  the  quantities  in  the  coefficient  of  61  are  not 
readily  determined,  it  is  simpler  to  express  this  quantity  in  terms 
of  the  figure  of  merit  of  the  galvanometer  and  its  period  of  oscilla- 
tion T.  Since  the  coil  executes  an  angular  harmonic  motion,  its 
period  is  given  by 


T  =  2-  (23) 

Substituting  from  (23)  and  (11)  in  (22)  there  results 

<2  =  g™  (24) 

The  constant  K  of  eq.  (15)  is  thus  seen,  for  the  undamped 


GALVANOMETERS 


27 


ballistic  galvanometer,  to  be  o~  times  its  figure  of  merit  when 

used  as  a  current  measuring  instrument. 

The  ideal  condition,  i.e.,  zero  damping,  cannot  be  realized  in 
practice.  Moreover,  it  would  be  exceedingly  cumbersome  to 
use,  because  of  the  difficulty  infringing  the  coil  to  zero  and  main- 
taining it  in  this  position  while  adjusting  other  parts  of  the  appa- 
ratus in  preparation  for  an  observation.  Since  a  certain  amount 
of  damping  must  necessarily  be  present,  it  is  usually  most  con- 
venient to  increase  the  damping  until  the  motion  is  just  aperiodic. 
In  this  case,  the  galvanometer  deflects  to  a  certain  point  and  then 
returns  to  zero  in  the  quickest  time;  and,  barring  external  dis- 
turbances, remains  in  this  position  indefinitely. 

The  theory  of  the  damped  ballistic1  galvanometer  is  somewhat 
involved  and  is  beyond  the  scope  of  this  book.  It  may  be  shown, 
however,  that  when  damping  exists,  the  quantity  of  electricity 
passed  through  is  given  by 

o-sX'  +  D*  (25) 

where  X  is  called  the  "  logarithmic  decrement"  and  is  defined  as 

the  Naperian  logarithm  of  the 

ratio  of  any  deflection  to  the 

next  one  succeeding  it  in  the 

same   direction.      It  is  thus 

seen  that  damping  reduces  the 

ballistic  sensitivity  of  a  gal- 

vanometer.     Further,  if  the 

galvanometer  is  standardized 

under    conditions    such    that 

the  damping  is  different  from 

what  it  is  in  use,  the  decre- 

ment must  be  determined  in 

both  cases  and  the  difference 

allowed  for  by  eq.  (25). 

27.  Determination  of  the 
Constant  of  a  Ballistic  Gal- 
vanometer. —  The  standardi- 


FIG. 9. — Condenser  and  standard  cell 
method  for  obtaining  constant  of  ballistic 
galvanometer. 


zation   of  a  ballistic  galvanometer  consists  in  passing  known 
quantities  of  electricity  through  it  and  measuring  the  deflec- 
i  O.  M.  STEWART,  Phya.  Rev.,  vol.  XVI,  1903,  p.  158. 
LAWS,  Electrical  Measurements,  chap.  II. 


28  ELECTRICITY  AND  MAGNETISM 

tions  they  produce.  Two  methods  are  in  common  use,  known 
respectively  as  the  "  condenser  and  standard  cell  method"  and 
the  mutual  inductance  or  "  standard  solenoid  method." 

1.  The  Condenser  and  Standard  Cell  Method.  —  This  method 
consists  in  charging  a  condenser  of  known  capacity  by  means  of 
a  standard  cell,  and  then  discharging  this  quantity  through  the 
galvanometer.  The  apparatus  is  arranged  as  shown  in  Fig.  9, 
where  G  is  the  galvanometer  to  be  standardized,  C  a  standard 
condenser,  K  a  charge  and  discharge  key,  and  S  a  standard  cell. 
If  V  is  the  E.  M.  F.  of  the  cell,  the  quantity  stored  in  the  con- 
denser when  the  key  is  pressed  down  is 

Q  =  CV  (26) 

and  since 

Q  =  Kd  (27) 

we  have 


If  C  is  a  subdivided  condenser,  several  different  values  should  be 
used,  a  curve  plotted  using  Q  as  abscissas  and  d  as  ordinates,  and 
the  constant  computed  from  the  slope  of  the  straight  line.  If  C 
is  expressed  in  farads,  V  in  volts,  and  d  in  centimeters,  K  will  be 
given  in  coulombs  per  centimeter;  but  if  C  is  in  micro-farads,  K 
will  be  given  in  micro-coulombs  per  centimeter. 

2.  The  Standard  Solenoid  Method.  —  This  method  is  especially 
applicable  to  cases  in  which  the  galvanometer  is  used  on  low 
resistance  circuits  where  the  damping  is  large.  The  known 
quantity  of  electricity  discharged  through  the  galvanometer  is 
obtained  from  the  secondary  of  a  standard  mutual  inductance 
when  a  measured  change  in  the  primary  current  is  produced. 
The  connections  are  shown  in  Fig.  10  where  AD  is  the  primary 
of  the  mutual  inductance,  SS'  the  secondary  coil,  and  G  the 
galvanometer  to  be  calibrated. 

Let  Q  =  quantity  of  electricity  discharged  through  the  galvano- 
meter. 

i  =  instantaneous  current  in  galvanometer. 
e  =  instantaneous  E.M.F.  in  secondary  coil. 
/  =  value  of  primary  current. 
R  =  total  resistance  of  secondary  circuit. 
M  —  mutual  inductance  between  AD  and  SS. 
T  =  time  required  for  discharge  to  take  place. 


GALVANOMETERS 
Then,  from  the  above, 

Q  =  Kd 
But 


, 
= 


T 

idt 


e  dl 

i  =  -=  and  e  —  M  —^  from  definition 
ti  at 


Hence, 


Kd 


TMdI,.  _  M  C1 ,r  _  MI 

~~^.  ~rrCLt   —  ~^r  I    dl    —      7^ 
£1 


or 


„      Mil 
K~~Rd 


29 

(29) 
(30) 
(31) 
(32) 


I  ill  ll VWWWW 


FIG.  10. — Standard  solenoid  method  for  ballistic  galvanometer  constant. 

If  one  of  the  coils  is  uniformly  wound  and  has  a  length  great  in 
comparison  to  its  diameter,  as  the  primary  AD  of  Fig.  10,  it  is 
called  a  standard  solenoid.  The  mutual  inductance  may  then 
be  calculated  from  the  dimensions  of  the  solenoid,  and  the 
number  of  turns  on  the  coils,  as  follows : 

Let  A  =  area  of  standard  solenoid 
L  =  length  of  standard  solenoid 
N  =  number  of  turns  on  standard  solenoid 
H  =  field  strength  in  standard  solenoid 
$  =  total  flux  in  standard  solenoid 
n  =  turns  on  secondary  of  standard  solenoid. 


30  ELECTRICITY  AND  MAGNETISM 

The  coefficient  of  mutual  inductance  may  be  defined,  in  electro- 
magnetic units,  as  the  number  of  magnetic  linkages  through  the 
secondary  when  unit  current  is  flowing  in  the  primary,  where,  by 
linkages  is  understood  the  product  of  the  number  of  turns  and  the 
total  flux.  As  the  secondary  coil  surrounds  the  standard  solenoid, 
we  have 

M  =  rc</>  =  nHA  (33) 

4.irNnA     ,  ,.  ,_.. 

=  —  j  —  electromagnetic  units  (34) 

Li 

Since,  however,  we  wish  M  expressed  in  henries,  we  must  divide 
by  109,  the  number  of  E.M.U's.  required  for  one  henry.  Accord- 
ingly, our  equation  becomes 

4irNnA  I 
"- 


It  is  customary  to  reverse  the  current  through  the  primary  of  the 
standard  solenoid  instead  of  merely  "making"  it  as  implied  in  the 
above  derivation.  The  limits  of  integration  in  equation  (31) 
should  then  be  —7  and  +7  instead  of  0  and  7,  in  which  case 
our  formula  becomes 

SirNnA  I  . 

"- 


In  the  above  derivation,  we  have  assumed  that  the  field 
strength  at  the  center  of  the  standard  solenoid  is  given  by  the 
formula 


_  (     } 

~ 


which  is  true  only  for  an  infinitely  long  solenoid.  If  the  length 
of  the  standard  solenoid  is  fifty  times  the  diameter,  the  error, 
which  is  due  to  the  demagnetizing  effects  of  the  ends,  is  less  than 
one-half  of  one  per  cent.  We  have  further  assumed  that  there  is 
no  magnetic  leakage  between  primary  and  secondary  coils,  a 
condition  which  is  never  realized.  Our  value  for  M  ,  computed 
above,  is,  therefore,  too  large;  and  for  very  accurate  work,  a 
correction  should  be  made.  If  we  call/  the  demagnetization  and 
leakage  factor,  our  corrected  formula  for  K  becomes 


In  practice,  it  is  customary  to  obtain  a  series  of  deflections 
using  different  values  of  7,  then  plot  7  as  abscissas  and  d  as 

ordinates,  and  obtain  the  ratio  -3-  from  the  slope  of  the  line.     If 


GAL  VA  NO  METERS 


31 


practical  units  of  electrial  quantities  are  used  throughout,  K  will 
be  expressed  in  coulombs  per  centimeter. 

28.  The  Fluxmeter. — It  was  pointed  out  above,  as  a  necessary 
condition  that  the  ballistic  galvanometer  should  give  indications 
proportional  to  the  quantity  of  electricity  passed  through  it,  that 
this  passage  must  be  completed  before  the  moving  system  swings 


FIG.  11. — Grassot  Flux  Meter. 

appreciably  from  its  zero  position.  In  certain  instances,  as, 
for  example,  the  testing  of  iron  possessing  magnetic  viscosity, 
the  induced  current  which  is  passed  through  the  galvanometer 
persists  too  long,  and  hence  the  ordinary  instrument  cannot  be 
used.  The  Grassot  fluxmeter  is  a  modified  ballistic  galvano- 
meter of  the  moving  coil  type,  in  which  this  difficulty  is  over- 
come. The  coil  is  suspended  by  a  fine  silk  fibre  and  is  practically 
free  from  restoring  forces,  the  current  being  led  in  and  out  by 


32  ELECTRICITY  AND  MAGNETISM 

means  of  fine  helical  springs.  It  is  rectangular  in  shape  and  is 
placed  in  a  field  as  nearly  radial  as  possible,  with  respect  to  its 
axis  of  rotation,  the  parts  involved  being  similar  to  those  of  the 
Weston  ammeter.  The  torque,  for  a  given  current,  is  practi- 
cally independent  of  the  position  of  the  coil.  When  connected 
to  a  resistance  equal  to  or  less  than  its  critical  resistance,  the 
coil  is  stationary  in  any  position.  When  a  given  quantity  of 
electricity  is  discharged  through  it,  it  moves  to  a  new  position  and 
the  change  in  position  is  proportional  to  the  quantity  that 
passed,  no  matter  how  long  a  time  was  required.  It  is  standard- 
ized and  used  as  an  ordinary  ballistic  galvanometer,  except  that 
some  means  must  be  provided  for  bringing  it  back  to  its  zero 
position.  Figure  11  shows  the  construction  of  an  instrument  of 
this  type. 

29.  Theory  of  the  Fluxmeter.1 — As  originally  designed,  the 
fluxmeter  was  intended  as  an  instrument  for  the  direct  measure- 
ment of  magnetic  flux  density.  For  this  purpose,  coils  are  con- 
structed which  consist  of  a  definite  number  of  turns  wound  on  a 
plate  of  nonmagnetic  material,  the  area  of  which  must  be  care- 
fully measured.  These  coils  are  made  very  thin  so  that  they  may 
be  inserted  in  a  narrow  air  gap  such  as  exists  between  the  arma- 
ture and  pole  pieces  of  a  dynamo.  The  measurement  of  an  un- 
known flux  density  consists  then  in  connecting  the  test  coil  by  flex- 
ible leads  directly  to  the  fluxmeter  and  placing  it  at  right  angles  to 
the  flux  to  be  measured.  The  instrument  is  brought  to  zero  by 
some  suitable  device.  The  test  coil  is  then  withdrawn  from  the 
flux  and  the  accompanying  deflection  of  the  instrument,  multiplied 
by  its  constant,  measures  directly  the  change  in  flux  through  the 
test  coil. 

The  direct  proportionality  between  change  of  flux  through  the 
coil  and  deflection  of  the  instrument  may  be  shown  as  follows: 

Let  0  =  flux  through  the  exploring  coil 
N  =  number  of  turns  in  exploring  coil 
L  =  inductance  of  exploring  and  galvanometer  coils 
R  =  resistance  of  exploring  and  galvanometer  coils 
C  =  constant  of  galvanometer  coil  =  Hnlb 
I  =  moment  of  inertia  of  galvanometer  coil 

1  LAWS,  Electrical  Measurements,  p.  124. 

M.  E.  GRASSOT,  Fluxmetre,  Journal  de  Physique,  4th  series,  vol.  3,  1904, 
p.  696. 


GALVANOMETERS  33 

(a  =  angular  velocity  of  galvanometer  coil 

i  =  instantaneous  current  in  galvanometer  coil 

8  =  angular  deflection  due  to  change  of  flux 

As  the  test  coil  is  withdrawn  from  the  flux,  there  is  induced  in 
it  an  E.M.F.  given  by  ^-*    This    is   opposed   by  the  counter 

E.M.F.,  L  -j-.'  due  to  the  inductance  of  the  galvanometer  coil,  and 

also  by  an  E.M.F.  Ceo  due  to  the  motion  of  this  coil  through  the 
field  of  the  instrument.  Accordingly,  the  current  at  any  instant 
is 


Ar  d<f>       T  di       n 
N  j-  --  L  -j.  —  Ceo 

_*__*  — 


(39) 


The  motion  of  the  coil  is  given  by 

n-  _    T  <**>  _  CN  d0       CL  di      C2co 

1  di  "     R    dt  '  "7f  dt  "  IT 

Integrating  between  the  limits  o  and  t,  where  t  is  the  duration  of 
the  change  of  flux  and  consequent  motion  of  the  galvanometer 
coil,  we  have 

CN  fd0,,       T   fdw  ,.  LCL  f<&-      ^ 

-R  Jo**  =  7  JodT  d<  +1F  J  **  + 

Remembering  that  at  both  limits  the  current  and  angular  velocity 
are  each  zero  and  that  /*co  dt  =  6,  we  have 

02  -  01  =  cNe  (42) 

The  change  of  flux  through  the  test  coil  is  thus  seen  to  be 
directly  proportional  to  the  angle  6  through  which  the  coil  rotates. 
This  deflection  may  be  read  either  by  a  pointer  or  a  mirror  and 
scale.  The  fluxmeter  may  be  used  for  almost  any  purpose  for 
which  the  ballistic  galvanometer  is  suited,  but  has,  in  general,  a 
somewhat  lower  sensitivity. 

30.  Checking  Devices.  —  If  a  ballistic  galvanometer  is  not 
critically  damped,  it  is  convenient  to  have  some  device  to  check 
its  motion  and  to  set  it  accurately  at  its  zero  position.  If  the 
instrument  is  of  the  D'Arsonval  type,  this  may  usually  be  accom- 
plished simply  by  a  short  circuiting  key.  However,  since  most 
keys  possess  slight  thermal  E.M.F.'s,  the  zero  with  the  key 
closed  will  usually  be  different  from  the  normal  zero  with  the  key 
open.  When  the  galvanometer  is  used  on  a  closed  circuit,  the 


34 


ELECTRICITY  AND  MAGNETISM 


device  shown  in  Fig.  12  is  much  more  satisfactory.  It  consists  of 
a  coil  of  wire  through  which  a  bar  magnet  may  be  moved.  The 
coil  is  connected  in  series  with  the  galvanometer  and  the  motion  of 
the  magnet  induces  in  it  a  small  E.M.F.,  positive  or  negative, 


Ballistic 
Galvanometer  I 


To  Apparatus 


£J 


FIG.  12. — Checking  device  for  ballistic  galvanometer. 

depending  upon  the  direction  of  motion.  The  key  must  remain 
closed  except  when  it  is  necessary  to  "get  a  new  hold"  on  the 
galvanometer.  With  a  little  experience  the  instrument  may, 
with  this  device,  be  set  on  zero  very  quickly  and  accurately. 


CHAPTER  III 
MEASUREMENT,  OF  RESISTANCE 

31.  Ohm's  Law. — When  a  current  of  electricity  is  flowing  from 
one  point  to  another  along  a  conductor,  a  difference  of  potential 
is  found  to  exist  between  these  points.     The  magnitude  of  the 
difference  of  potential  depends  upon  the  current  and  upon  a 
property  of  the  material  in  virtue  of  which  it  offers  opposition  to 
the  passage  of  current.     The  relation  between  potential  difference 
and  current  was  first  given  by  Ohm,  and  is  known  as  Ohm's  law. 
It  states  that,  as  long  as  the  physical  condition  of  a  conductor 
remains  unchanged,  there  is  a  constant  ratio  between  the  current 
and  potential  difference;  or,  in  symbols, 

'-I  <» 

where  the  proportionality  factor  R  is  called  the  resistance  of  the 
conductor.  This  law  is  a  result  of  experiment  and  has  been  found 
to  be  true  within  the  limits  of  the  most  refined  measurements. 

32.  Specific  Resistance. — For  a  uniform  conductor,  other  con- 
ditions remaining  the  same,  the  resistance  is  proportional  to  the 
length  and  inversely  proportional  to  the  area  of  cross  section. 
Hence,  if  I  represents  the  length  and  a  the  cross  section,  we  have 

R  =  P[  (2) 

where  p  is  a  constant  depending  upon  the  material  of  the  conduc- 
tor. Considering  this  as  a  defining  equation  for  p,  we  see  that, 
when  I  and  a  are  unity,  p  equals  R.  The  constant  p  is  thus  the 
resistance  of  a  unit  cube  of  the  material,  and  is  known  as  the 
Specific  Resistance.  In  tabulating  the  resistivities  of  substances, 
the  specific  resistance  is  a  convenient  quantity  to  use,  since 
knowing  it,  one  can  readily  compute  the  resistance  of  a  conductor 
of  any  length  and  cross  section  by  means  of  eq.  (2).  The 
value  of  p  depends  upon  the  units  employed  for  the  measurement 
of  length  and  resistance.  Since  the  resistance  of  a  unit  cube  of 
any  metal  is  a  very  small  quantity,  it  is  customary  to  express  the 
specific  resistance  in  microhms  per  centimeter  cube  where  a 

35 


36  ELECTRICITY  AND  MAGNETISM 

microhm  is  one  millionth  of  an  ohm.  Alloys,  in  general,  have  a 
much  higher  specific  resistance  than  pure  metals,  and  the  pres- 
ence of  even  a  trace  of  another  metal  which,  of  itself,  may  be  a 
good  conductor,  has  a  considerable  effect  upon  the  resistance;  and 
hence,  copper,  for  electrical  purposes,  should  be  pure. 

33.  Temperature  Coefficient  of  Resistance. — The  resistance  of 
all  conductors  is  found  to  change  with  the  temperature.     In  the 
case  of  the  pure  metals,  the  resistance  increases  with  increasing 
temperature,  while  for  carbon  and  electrolytes,  the  opposite  is 
true.     The  former  are  said  to  have  a  positive,  and  the  latter  a 
negative,  temperature  coefficient.     Experiment  shows  that,  over 
relatively  large  intervals  of  temperature,  the  resistance  of  a  given 
conductor,   at  any  temperature   t,   may  be  expressed  by    the 
equation 

Rt  =  R0(\  +  at  +  (3t*+ )  (3) 

where  R0  is  the  resistance  at  zero  degrees  and  a  and  /3  are  constants 
depending  upon  the  material  and  the  temperature  interval  con- 
cerned. Over  small  ranges  of  temperature,  the  change  in  resis- 
tance is  nearly  proportional  to  the  change  in  temperature,  and 
may  be  represented  by  the  linear  relation 

Rt  =  R0(l  +  at).  (4) 

The  coefficient  a  is  called  the  "  Temperature  Coefficient,"  and  is 
the  change  in  resistance  per  ohm  per  degree  change  in  tempera- 
ture. Some  alloys,  such  as  german  silver  and  manganin,  have  a 
very  small  temperature  coefficient,  that  of  the  latter  being  zero  at 
some  temperatures.  Manganin  is  well  suited,  for  this  reason, 
for  the  construction  of  standard  resistances. 

34.  Measurement  of  Resistance. — The  independent  or  "  abso- 
lute" determination  of  resistance,  that  is,  measurement  in  terms  of 
the  fundamental  units  of  length,  mass,  and  time,  is  a  matter  of 
considerable   difficulty;   and   so  the   establishment   of   primary 
standards  is,  at  the  present  time,  left  almost  entirely  to  govern- 
ment Bureaus  of  Standards,  which  are  especially  equipped  for  work 
of  this  character.     On  the  other  hand,  the  comparison  of  resis- 
tances, even  to  a  high  degree  of  accuracy,  is  relatively  simple,  and 
it  is  with  work  of  this  character  only  that  we  are  concerned  here. 

35.  The    Wheatstone    Bridge. — This    is    the   usual   method 
employed  for   comparing   resistances   of   ordinary  magnitudes, 
and  its  principle  may  be  readily  understood  from  Fig.  13.     Four 
resistances  are  connected  in  the  form  of  a  diamond,  with  current 


MEASUREMENT  OF  RESISTANCE 


37 


from  the  battery  entering  at  A,  where  it  divides  in  two  parts 
which  unite  again  at  B.  The  galvanometer  G  is  connected  across 
the  other  corners  of  the  diamond. 

Since  the  points  P  and  Q  possess  potentials  intermediate  between 
those  of  A  and  B,  it  must  be  possible  to  make  Q  have  the  same 
potential  as  P  by  suitably  choosing  Rs  and  7£4.  When  this 
condition  has  been  established, .  no  current  flows  through  the 
galvanometer,  as  indicated  by  zero  deflection,  and  the  bridge  is 


FIG.   13.— Wheatstone  Bridge. 

said  to  be  balanced.     Calling  the  current  through  Ri  and  R%, 
Ci,  and  that  through  Rs  and  R*,  C^  we  have  the 

P.D.  between  A  and  P  =  P.D.  between  A  and  Q  and 
P.D.  between  P  and  B  =  P.D.  between  Q  and  B. 
By  Ohm's  law 

Rid  =  RSC,  (5) 

and 

R*Ci  =  R^Cz  (6) 

Whence 

-1  =  —  m 

R2       R* 

This  is  the  law  of  the  Wheatstone  bridge,  and  it  is  clear  that,  if 

three  of  these  resistances  are  known  the  fourth  may  be  computed. 

36.  The  Slide  Wire  Bridge. — If,  in  the  above  equation,  Rl 

is  an  unknown  and  R%  a  standard  resistance,  the  former  may  be 


38 


ELECTRICITY  AND  MAGNETISM 


expressed  in  terms  of  the  latter  by  means  of  the  ratio  of  Rs  to  #4, 
It  is  obvious  then  that  the  actual  values  of  Rz  and  R±  need  not  be 
known,  their 'ratio  being  sufficient.  Advantage  is  taken  of  this 
fact  in  the  construction  of  the  slide  wire  bridge,  which  is  shown 
diagrammatically  in  Fig.  14  where  the  corresponding  points  of 
Fig.  13  are  indicated  by  the  same  letters.  R%  and  R±  are  replaced 


FIG.  14. — Slide  wire  bridge. 

by  portions  of  the  slide  wire  SW  and  their  magnitudes  varied  by 
moving  the  slider  Q.  Calling  p  the  resistance  of  1  cm.  of  the  wire, 
we  have 

X  =  pi, 

R       plz 
whence 


(8) 


X  =  fR. 


(9) 


In  order  to  increase  the  accuracy  of  setting,  and  to  reduce  the 
relative  errors  in  measuring  li  and  1%,  especially  where  X  and  R 
have  quite  different  values,  resistances  are  introduced  in  place 
of  the  links  M  and  AT,  which  may  be  measured  in  terms  of  p  and 
expressed,  therefore,  as  a  certain  number  of  slide  wire  units  to  be 
added  to  li  and  h. 

37.  The  Post-office  Box. — A  more  compact  form  of  Wheat- 
stone  bridge  is  shown  in  Fig.  15,  which  is  known  as  the  post-office 
box,  from  the  fact  that  it  was  adopted  at  an  early  date  by  the 
British  Post  and  Telegraph  Office.  The  slide  wire  is  replaced  by 
two  series  of  ratio  coils,  AQ  and  BQ,  having  resistances  of  10, 
100,  1,000  and  10,000  ohms  each,  while  the  third  arm  is  a  series 


MEASUREMENT  OF  RESISTANCE 


39 


of  coils,  arranged  as  in  the  ordinary  resistance  box,  frequently 
having  a  total  of  100,000  ohms.  The  unknown  X  is  connected 
between  B  and  P.  Since  the  ratio  of  X  to  R  is  thus  a  decimal 
number,  no  calculation  is  required.  With  the  ratio  coils  set  at 

5 


CX     () 

O      O 

o 

1P   O        C)       CL 

0 

°l 

IV/0000 

1000                100 

10 

1  .-,      10                    IOO                 IOOO 

LT 

(0000 

1 

\ 

1 

O 

o 

\  0 

<!>  |    (•:> 

C) 

1 

a 

1     2 

1 

j 

*l     * 

20 

n 

1    _ 

O 

!  0 

()   ,        O 

() 

1 

**w 

Soo 

I      200 

200     |              too 

\ 

5* 

1 

Q 

QE 

O  i       C) 

O 

°i 

5000 


FIG.  15. — Post-Office  box  diagram. 

1,000:1,  resistances  up  to  100  megohms  may  be  measured;  while 
with  the  ratio  reversed,  resistances  of  the  order  of  .001  may  be 
detected.  The  range  is  thus  great  and  its  advantages  are  obvious. 
In  using  the  box  bridge,  one  should  first  use  a  1:1  ratio,  setting 
the  coils  at  100  ohms  each,  and  obtain  a  rough  balance,  thus  finding 


FIG.  16.— Post-office  box. 


the  order  of  magnitude  of  the  unknown.  He  should  then  choose 
such  a  ratio  as  will  cause  the  balance  setting  of  R  to  be  as  large 
as  possible.  For  example,  suppose  C  is  found  to  be  of  the  order  of 
45  ohms.  By  using  a  ratio  of  1 : 1,000  a  balance  may  be  obtained 


40 


"ELECTRICITY  AND  MAGNETISM 


at  45,638,  let  us  say,  giving,  as  the  value  of  the  unknown,  45,638; 
while  if  a  ratio  of  1 :100  had  been  used,  the  result  would  have  been 
45.64.  The  higher  ratio  thus  increases  the  accuracy.  Box  bridges 
of  the  better  class  are  provided  with  plugs  for  interchanging  the 
ratio  arms,  by  means  of  which  inequalities  in  the  internal  connec- 
tions of  the  bridge  may  be  eliminated,  and  a  check  obtained  upon 
the  accuracy  of  the  ratio  coils.  For  accurate  work,  one  should 
reverse  the  battery  terminals  in  each  case  and  re-balance,  thus 
eliminating  errors  due  to  thermal  and  contact  differences  of 
potential.  A  convenient  form  of  post-office  box  is  shown  in 
Fig.  16. 

38.  Measurement  of  Low  Resistance.1  Kelvin's  Double 
Bridge. — For  the  measurement  of  extremely  low  resistances 
such  as  that  of  a  few  feet  of  trolley  wire,  cable,  bus-bars,  etc., 
the  Wheatstone  bridge  is  unsuited  for  two  reasons:  First,  when 
the  resistances  to  be  compared  are  very  low,  the  bridge  becomes 
insensitive;  and  second,  some  sort  of  connectors  must  be  used  for 
A  x  B  joining  the  unknown  to  the 

bridge,  and  these  may  have  a 
resistance  comparable  to  that 
to  be  measured.  The  Kelvin 
double  bridge  avoids  both  of 
these  difficulties.  The  general 
scheme  of  this  circuit  is  shown 
in  Fig.  17,  where  X  and  S  are 
the  unknown  and  standard  re- 
sistances, respectively,  through 
which  a  large  current  flows 
which  need  not  necessarily  be 
constant.  There  are  four  ratio 

coils,  a,  b,  c,  and  d,  arranged  in  pairs,  while  the  galvanometer  is 
connected  at  the  points  C  and  D,  between  each  pair.  By 
properly  adjusting  the  ratio  coils,  C  and  D  may  be  brought  to 
the  same  potential,  when  no  current  flows  through  the  galvano- 
meter and  the  currents  in  X  and  S  are  equal.  When  the  balance 
has  thus  been  obtained,  let  us  call  I  the  current  through  X  and  S, 
1 1  that  through  a  and  c,  and  72  that  through  b  and  d.  Then,  by 
Ohm's  law, 

c/i  =  XI  +  dI2  and  al,  =  SI  +  b!2  (10) 

1  NORTHRUP,  Measurement  of  Resistance,  chap.  VI. 
LAWS,  Electrical  Measurements,  chap.  IV. 


F        s        b 

FIG.   17. — Diagram  for  Kelvin's 
double  bridge. 


MEASUREMENT  OF  RESISTANCE 


41 


Whence 


XI  = 
XI  = 


i  -  d!2  and  SI 


=  al^  -  bI 


By  the  construction  of  the  instrument, 

d  =  b 

c  &  a 

-«" 

which  gives,  on  dividing  equations  (12), 

X       c 

S  ~~    a 


(11) 
(12) 


(13) 


which  is  the  working  formula  for  the  instrument.  One  form  of 
this  bridge  devised  by  Leeds  and  Northrup,  is  shown  in  Fig.  18 
where  the  points  corresponding  to  those  of  the  schematic  diagram 


FIG.   18. — Laboratory  form  of  Kelvin's  double  bridge 

of  Fig.  17,  are  lettered  similarly.  The  unknown  is  represented 
as  a  heavy  rod  with  potential  taps  at  A  and  B,  while  the  standard 
consists  of  the  bar  MN  and  the  coils  with  posts  numbered  0-9. 
Each  of  the  coils,  as  well  as  the  standard  bar,  has  a  resistance  of 
.01  ohms,  and  the  resistance  being  used  as  the  standard  S,  is 
that  between  the  slider  F  and  the  movable  plug  E.  The  standard 
thus  has  a  range  of  0  to  .1  ohms  by  infinitesimal  steps.  The  ratio 
coils  are  situated  to  the  right  of  the  standard  coils  and  are 
connected  to  the  galvanometer  in  different  ways  by  means  of  the 
plugs  C  and  D.  A  little  study  of  the  connections  will  show  that 
three  different  ratios  are  possible;  namely,  1:10,  1:1,  and  10:1. 
The  plugs  C  and  D  must  be  placed  opposite  one  another,  since  a 
double  ratio  must  be  maintained  as  indicated  by  the  equations; 
that  is, 

-  =  -  =  d  (14) 

Sab 


42  ELECTRICITY  AND  MAGNETISM 

The  resistance  X  is  that  portion  of  the  rod  between  the  points  A 
and  B  only.  When  the  resistance  to  be  determined  is  of  some 
other  form  than  a  rod,  it  must  be  provided  with  two  sets  of  leads; 
a  heavy  pair  for  the  current,  which  are  joined  to  the  bridge  at  S 
and  T,  and  a  light  pair  for  the  potential  drop  across  it,  joined  at 
I  and  ra.  The  bridge  thus  measures  the  resistance  of  the  con- 
ductor between  the  points  to  which  the  potential  leads  are 
attached. 

39.  Experiment     1.     Specific    Resistance     of    Materials. — In 
this  experiment,  the  specific  resistance  of  three  metals,  copper, 
brass,  and  iron,  is  to  be  found.     The  metals  are  provided  in  the 
form  of  rods,  which  are  to  be  clamped  in  the  bridge  at  S  and  T. 
Make  sure  that  good  contact  is  obtained  at  A  and  B  by  polishing 
the  bars  at  those  points  with  emery  cloth.     Use,  as  a  current 
supply,  a  ten-volt  storage  battery  and  include  an  ammeter  and 
a  reversing  switch  in  this  circuit,  and  a  press  key  in  the  galvano- 
meter circuit.     Operate  the  bridge  on  3  amperes.     Measure  the 
resistance  of  50  cm.  and  100  cm.  lengths  of  each  bar,  reversing 
the  current  at  each  setting  to  eliminate  errors  due  to  thermal  and 
contact  potential  differences  within  the  instrument.     Make  at 
least  four  balances   for  each  length  approaching  the  balance 
point  from  both  sides.     Determine  the  diameter  of  the  rod  by 
means  of  a  micrometer  gauge,  taking  the  average  of  ten  measure- 
ments uniformly  distributed  over  its  length. 

Report. —  1.  Compute  the  specific  resistance  of  each  material 
in  michroms  per  centimeter  cube. 

2.  Compare  your  results  with  the  data  given  in  one  or  two  of 
the  standard  tables  of  physical  constants  to  be  found  on  the 
reference  shelves.  How  do  you  account  for  the  discrepancies? 

40.  Carey-Foster  Method  for  Comparing  Two  Nearly  Equal 
Resistances. — A    very    accurate    method    for    comparing   two 
resistances  which  are  nearly  equal  to  one  another  has  been 
devised  by   Carey  Foster.1     It  possesses  the   advantage  that 
errors  arising  from  the  resistance  of  leads  within  the  bridge,  as 
well  as  those  due  to  thermal  and  contact  electromotive  forces, 
providing  they  remain  constant,  are  automatically  eliminated. 
The  wiring  diagram  is  shown  in  Fig.  19.     It  consists  of  a  slide 
wire  bridge  in  which  RI  and  R2  are  ratio  coils  and  A  and  B  are 
the  resistances  to  be  compared.    Let  r\  and  r2  be  the  resistances 
of  the  internal  bridge  leads  between  the  battery  and  slide  wire 

1  Phil.  Mag.,  May,  1884. 


MEASUREMENT  OF  RESISTANCE 


43 


connections  on  each  side.     Then  if 
bridge  wire  at  balance,  we  have 

Ri        A_+ri 

5i  = 


and  mx  are  lengths  of  the 


+ 


(15) 


£+r2  + 

where  p  is  the  resistance  of  the  slide  wire  per  unit  length.  Now 
let  A  and  B  exchange  places,  and  let  Z2  and  m2  be  the  correspond- 
ing lengths  for  a  new  balance.  Then 


r2  -f 


(16) 


"11  — 

fll 

#2 

B 

• 

. 

••    J> 

*-2 

I  m 

FIG.   19. — Wiring  diagram  for  Carey-Foster  bridge. 

Equating  the  right  hand  members  of  equations  (15)  and  (16)  and 
taking  the  resulting  equation  by  addition,  we  have 

A  +  TI  +  pi i  -f  B  +  r2  + 


B  +  r2 


+ 


+ 


+ 


r2 


(17) 


Since  Zi  +  mi  =  /2  -f  wi2,  the  numerators  of  these  fractions  are 
equal;  the  denominators  are  therefore  also  equal,  whence 

B  +  r2  +  pwi  =  A  +  r2  +  pm2 
A  -  B  =  P(W!  -  w2)  =  P(Z2  -  h)  (18) 

The  difference  in  the  resistance  of  the  two  coils,  A  and  B,  is  thus 
seen  to  be  equal  to  the  resistance  of  the  slide  wire  between  the 
two  points  of  balance,  before  and  after  the  interchange  of  the 
coils.  It  is  to  be  noted  that  this  result  is  independent  of 
the  values  of  R i  and  R<t 


44 


ELECTRICITY  AND  MAGNETISM 


The  Carey-Foster  bridge  is  usually  employed  for  the  com- 
parison of  coils  whose  temperature  must  be  maintained  constant 
and  they  are  usually  immersed  in  some  sort  of  oil  bath  for  this 


FIG.  20. — Coil  interchanger  for  Carey-Foster  bridge. 

purpose.  A  convenient  device  therefore  must  be  provided  for 
interchanging  them  without  removing  them  from  their  baths  or 
producing  any  changes  in  contact  resistances  by  handling  them. 


FIG.  21. — Complete  Carey-Foster  bridge. 


Figure  20  shows  an  arrangement  for  this  purpose.  The  coils  are 
supported  at  the  ends  of  heavy  copper  bars  which  swing  so  as  to 
receive  units  of  different  sizes.  Contact  between  resistance 


MEASUREMENT  OF  RESISTANCE  45 

terminals  and  bars  as  well  as  between  bars  and  links  of  the  com- 
mutator are  made  by  boring  cups  in  the  bars  and  partially  filling 
them  with  mercury.  The  interchange  of  the  coils  is  effected  by 
a  half  turn  of  the  commutator  at  the  center.  Adjustable  legs 
enable  the  coils  to  be  lowered  in  the  baths  to  the  proper  depth. 

To  adapt  the  bridge  to  the  comparison  of  coils  of  high  as  well 
as  low  resistances  and  to  securest  the  same  time  a  satisfactory 
sensitivity,  it  is  important  to  have  several  slide  wires  of  different 
resistances  per  unit  length.  Figure  21  shows  a  complete  bridge 
in  which  any  one  of  three  slide  wires  may  be  used  at  will.  To 
obtain  the  effect  of  a  very  low  resistance  slide  wire,  one  of  ordi- 
nary magnitude  may  be  shunted.  A  link,  seen  at  the  front  of  the 
switch  board  is  provided  for  this  purpose. 

41.  Determination  of  p. — The  Carey-Foster  method  requires 
that  the  slide  wire  be  of  uniform  resistance,  and  that  its  resistance 
per  unit  length  be  accurately  known.     To  measure  p,  the  process 
of  measurement  above  described  may  be  inverted,  using  for  A 
and  B  two  equal  coils  of  known  resistance,  one  of  which  is  shunted 
by   a   known   variable    resistance.     By    choosing    appropriate 
values  for  the  shunt,  any  desired  difference  between  A  and  B  may 
be  obtained,  and  by  changing  Ri  and  R%  the  balance  points  may 
be  shifted  to  different  positions  along  the  slide  wire.     The  con- 
stant p  is  obtained  by  substituting  in  eq.  (18). 

42.  Experiment  2.     Measurement  of  Temperature  Coefficient. — 
The  Carey-Foster   method  is  particularly  well  adapted  to  the 
measurement  of  the  variation  of  a  resistance  with  temperature. 
The  process  consists  in  determining  the  difference  between  the 
resistances  of  two  coils  one  of  which  is  constant  while  the  other  is 
changed  by  holding  it  at  different  temperatures.     The  metal, 
whose  coefficient  is  to  be  measured,  is  in  the  form   of  a  wire 
wound  on  a  frame  which  may  be  placed  in  an  oil  bath  to  secure  a 
uniform  temperature  throughout.     Place  the  container  in  an  ice 
pack  and  measure  the  resistance  at  a  temperature  as  near  as 
possible   to   0°C.     Next    place  the  container  in  a  water  bath 
heated   by   an  electric  heater.     Obtain  the   resistance   at    10° 
intervals  up  to  80°  C.     While  each  measurement  is  in  progress, 
remove     the    heater    and    place    the    bath    upon    a    wooden 
stand.     Stir  the  oil  continuously  and  read  t"he  thermometer  fre- 
quently.    Settings  should  be  made  as  rapidly  as  possible  to  avoid 
temperature  changes.     After  the  highest  temperature  has  been 
reached,  allow  the.  bath  to  cool  and  check  the  readings  at  three 


46  ELECTRICITY  AND  MAGNETISM 

points  on  the  way  down.     The  standard  resistance  should  also 
be  placed  in  an  oil  bath  and  its  temperature  maintained  constant. 

Report.  —  1.  Plot  a  resistance  temperature  curve  using  resis- 
tance as  ordinates  and  temperature  as  abscissas.  Draw  a 
straight  line  through  these  points  to  strike  an  average,  and  from 
it  determine  the  values  for  RQ  and  Rs$.  Compute  the  tempera- 
ture coefficient  a  from  the  equation 

Rt  =  RQ(1  +  at) 

2.  Consult  a  table  of  physical  constants  and  see  if  you  can 
identify  the  wire  tested  from  the  value  of  the  temperature  coeffi- 
cient obtained. 

43.  The  Measurement  of  High  Resistance.1  —  In  previous 
sections,  methods  for  measuring  resistances  of  ordinary  magni- 
tude and  for  very  small  resistances  have  been  considered.  The 
measurement  of  very  high  resistances,  such  as  the  insulation 
between  the  bus-bars  of  a  switch  board  and  the  ground,  the 
armature  bars  and  core  of  an  electrical  machine,  insulation  of 
cables,  etc.,  requires  special  consideration.  A  ready  method 
commonly  employed  by  engineers,  which  gives  reliable  results 
for  resistances  up  to  several  tenths  of  a  megohm,  and  even 
higher,  is  that  in  which  a  voltmeter  of  known  resistance  is  em- 
ployed, the  unknown  high  resistance  taking  the  place  of  the  multi- 
plier in  the  ordinary  use  of  the  instrument.  Suppose  a 
voltmeter,  of  resistance  r,  is  connected  across  a  source  of  E.M.F., 
and  the  voltage,  which  we  will  call  V,  is  measured.  Then  let 
an  unknown  resistance  R  be  connected  in  series  with  the  instru- 
ment across  the  same  source.  Since  the  voltmeter  measures 
the  fall  of  potential  across  its  own  internal  resistance,  which  we 
will  call  Vrj  while  the  total  voltage  across  R  and  r  is  that  origin- 
ally measured,  i.e.,  7,  we  may  write,  by  Ohm's  law, 


Or, 

.     '     ^  -  1 

Whence 


1  NORTHRUP,  Measurement  of  Resistance,  chap.  VIII. 
CARHART  and  PATTERSON,  Electrical  Measurements,  p.  92. 
GRAY'S,  Absolute  Measurements  in  Electricity  and  Magnetism,  p.  253. 


MEASUREMENT  OF  RESISTANCE 


47 


If  the  resistance  R  is  too  large,  Vr  will  be  insignificant  com- 
pared with  V,  and  the  method  evidently  will  not  yield  satisfactory 
results.  For  resistances  of  such  magnitude,  e.g.,  several  meg- 
ohms, recourse  is  generally  taken  to  some  leakage  method  in 
which  the  high  resistance  is  used  as  an  insulator,  and  its  magni- 
tude estimated  from  the  rate » at  which  a  known  charge  leaks 
through  it.  As  an  example,  consider  the  case  in  which  it  is 


L      I 

1 — vwvww—1 


FIG.  22. — Insulation  resistance  by  leakage. 


desired  to  measure  the  resistance  of  the  insulation  of  a  given 
length  of  cable.  The  cable  should  be  coiled  up  and  placed  in  a 
tank  of  water,  both  ends  being  left  outside.  This  arrangement 
may  be  considered  a  condenser,  one  plate  of  which  is  the  water, 
the  other  the  wire,  while  the  insulation  is  the  dielectric.  Its 
electrical  equivalent  is  shown  in  Fig.  22.  If  the  wire  and  water 
are  charged  to  a  given  potential  difference  and  the  insulation 
were  perfect,  the  charge  would  remain  constant;  but,  if  the  insu- 
lation possesses  a  slight  conductivity,  the  charge  will  gradually  leak 
through,  reducing  the  potential  difference  of  the  condenser. 
The  rate  of  leak  may  be  estimated  by  measuring  the  residual 
charge  after  leakage  has  been  going  on  for  a  definite  time  and 
comparing  it  with  the  original  charge.  The  resistance  is  then 
calculated  as  follows : 


48  ELECTRICITY  AND  MAGNETISM 

Let    C  =  capacity  of  the  coil 
V0  =    applied  voltage 
R  =  insulation  resistance  in  ohms 
F  =  instantaneous  difference  of  potential 
I  =  instantaneous  leakage  current 
Q  —  instantaneous  charge 

The  charge  Q  is  given  by 

Q  =  CV  (22) 

and 

dQ  _         dV       V 
~dt   '        Ldt  -R 
or 


Separating  the  variables 

f  +<!=«:,;      ; 

Integrating 

log,  F  -4-  JL  =  K  (26) 

where  K  is  a  constant  of  integration  to  be  determined  from  the 
initial  conditions.  For  this  purpose,  reckoning  time  from  the 
instant  when  the  leakage  begins,  the  condition  to  be  satisfied 
by  the  equation  is,  when  t  =  Q,  V  =  V0.  Substituting  these 
values  in  eq.  (26),  we  have  log  V0  =  K.  Replacing  K  by  this 
value,  we  have 

log,  V  +  ^  =  loge  V0  (27) 

or 

loge  V0  -  log,  V  =  -±  (28) 

Thus, 

**T-c8  (29) 

Solving, 


44.  Experiment  3.  Insulation  Resistance  by  Leakage. — Con- 
nect the  apparatus  as  shown  in  Fig.  22  where  G  is  a  ballistic 
galvanometer,  C  the  coil  under  test,  B  a  storage  battery,  and  K 
a  well  insulated  charge  and  discharge  key.  B  should  have  such 


MEASUREMENT  OF  RESISTANCE  49 

a  voltage  that  the  first  throw  of  the  galvanometer  is  about  15 
cms.  Since  this  is  a  leakage  experiment,  its  success  depends  upon 
having  all  parts  well  insulated;  the  tank  should  be  placed  upon  a 
glass  plate  or  an  insulated  stand,  and  care  be  taken  that  no  wires 
touch  each  other,  the  table,  or  other  apparatus.  Since  the 
capacity  of  the  coil  does  not  change  with  time,  the  deflections 
of  the  galvanometer  are  proportional  to  the  voltage  across  its 
terminals.  Charge  and^discharge  immediately  thus  obtaining  a 
deflection  proportional  to  V0.  Repeat  this  several  times  and  take 
the  average.  Then  charge  and  place  the  key  on  the  point  marked 
"  Insulate"  and,  after  allowing  15  seconds  for  leakage,  again 
discharge  and  obtain  a  deflection  proportional  to  'F.  Repeat 
the  operation  for  the  following  times  of  leak:  0.5,  1,  2,  5,  7,  10,  20, 
30  minutes.  If,  in  computing  the  resistance,  common  logarithms 
are  used,  the  modulus  for  changing  to  natural  logarithms  must  be 
introduced.  If  t  is  in  seconds,  and  C  in  farads,  R  will  be  expressed 
in  ohms;  but  if  C  is  in  microfarads  R  will  be  in  megohms.  The 
formula  becomes 

R  =  —  --  j  (31) 

2.303  C  loio 


The  capacitance  C  of  the  cable  may  be  obtained  from  the  relation 

Q0  =  CF0  =  kd0  (32) 

where  k,  the  constant  of  the  galvanometer,  is  to  be  obtained  by 
charging  a  standard  condenser  with  a  standard  cell  and  dis- 
charging through  the  galvanometer,  as  explained  in  Art.  27. 
Measure  VQ  by  an  ordinary  voltmeter.  Measure  the  resistance 
of  the  wire  of  the  cable  by  the  voltmeter-ammeter  method,  using 
for  this  purpose  about  20  amperes.  Determine  also  the  diameter 
of  the  wire  by  means  of  a  micrometer  gauge. 

Report.  —  1.  Compute  the  resistance  of  the  cable  for  each 
time  of  leak  from  eq.  31,  and  plot  the  insulation  resistance  in 
megohms  as  ordinates,  and  time  of  leak  as  abscissas.  It  will 
be  found  that  the  resistance  is  not  constant  but  increases  with 
the  time  during  which  it  was  subjected  to  a  voltage,  approaching 
assymptotically  to  a  limiting  value.  This  is  characteristic  of 
all  insulators  of  this  class,  and,  in  stating  their  resistances, 
the  time  for  which  it  was  determined  must  always  be  specified. 

2.  From  the  data  on  the  resistance  and  diameter  of  the  wire, 
find,  by  means  of  a  wire  table,  the  length  of  the  cable,  and  com- 


4 


50  ELECTRICITY  AND  MAGNETISM 

pute  the  insulation  resistance  per  mile  for  some  selected  time  of 
leak.  In  making  this  calculation,  remember  that  the  insulation 
resistance  is  measured  in  the  direction  in  which  the  leakage 
current  flows,  namely,  radially  from  the  wire  to  the  outside,  and 
that  the  longer  the  cable  the  less  will  be  the  total  insulation 
resistance. 

45.  The  Internal  Resistance  of  Cells.1 — It  is  a  well-known 
experimental  fact  that  when  a  cell  is  delivering  current,  the 
E.M.F.  across  its  terminals  is  not  the  same  as  on  open  circuit  but 
changes  with  the  current,  being  less  the  larger  the  current.  This 
is  true  not  only  for  cells,  but  for  all  electrical  generators  contain- 
ing internal  resistance.  Let  a  cell,  having  an  E.M.F.  of  E  volts 
and  an  internal  resistance  of  r  ohms,  be  connected  to  an  external 
resistance  of  R  ohms,  and  let  I  be  the  amperes  flowing;  then  the 
rate  at  which  energy  is  delivered  by  the  cell  is  El  watts.  Since 
the  current  must  flow,  not  only  through  the  external  resistance, 
but  also  through  the  internal  resistance  of  the  cell,  this  energy 
will  be  consumed  by  both  of  these  resistances;  PR  watts  in  the 
former,  and  Pr  watts  in  the  latter.  Accordingly  we  have 

El  =  (R  +  r)P  (33) 

or 

E  =  RI  +  rl  (34) 

This  is  an  equation  of  E.M.F.'s  which  states  that  the  total 
E.M.F.  of  the  cell  is  equal  to  the  external  plus  the  internal 
potential  drops.  Putting  the  terminal  P.D.  equal  to  E',  we  have 

E  -  E'  =  rl  (35) 

from  which  it  is  seen  that  the  internal  resistance  may  be  com- 
puted if  E,  E',  and  I  are  known.  In  fact,  this  is  the  method 
generally  employed  for  cells  which  are  able  to  furnish  a  consider- 
able current  without  polarization;  for  example,  storage  cells. 
Suppose  such  a  cell,  whose  internal  resistance  is  to  be  measured  is 
connected  as  shown  in  Fig.  23,  where  AM,  R,  and  K  are  an 
ammeter,  rheostat,  and  key,  respectively.  Let  the  voltmeter 
(V.M.)  be  an  instrument  taking  no  current,  e.g.,  an  electrostatic 
voltmeter,  having  an  infinite  resistance.  When  K  is  open,  the 
voltmeter  registers  the  total  E.M.F.  of  the  cell  because  there 
is  no  fall  of  potential  across  r,  as  no  current  is  flowing.  When  K 
is  closed,  however,  the  voltmeter  registers,  not  the  total  E.M.F. 

1  NORTHRUP,  Measurement  of  Resistance,  chap.  XI. 

CARHART  and  PATTERSON,  Electrical  Measurements,  pp.  96-105. 


MEASUREMENT  OF  RESISTANCE 


51 


as  before,  but  the  terminal  P.D.  E'  =  E  —  rl,  the  portion  rl 
being  consumed  in  sending  the  current  through  r.  Reading  now 
the  current,  the  internal  resistance  may  be  computed  from  Eq. 
35.  In  practice,  an  ordinary  Weston  voltmeter  may  be  used 
without  appreciable  error  since  the  resistance  of  the  voltmeter 
is  very  large  compared  with  that  of  the  cell,  and  the  ir  drop 
within  the  cell,  which  is  the  quaritity  by  which  the  indications 
of  the  instrument  differ  from  the  total  E.M.F.,  is  so  small  that  it 
may  be  neglected,  i  being  the  current  taken  by  the  voltmeter. 


V.M. 


A.M. 


FIG.  23. — Voltmeter-ammeter  method  for  internal  resistance  of  cells. 


If,  however,  the  cell  is  one  that  polarizes  rapidly,  this  method 
cannot  be  used,  since  E  and  E'  will  depend  upon  how  long  the 
current  has  been  flowing.  This  difficulty  may  be  overcome  by 
using  a  known  resistance  R  and  taking  the  voltages  so  quickly 
that  little  or  no  polarization  sets  in.  The  current  /  is  given  by 

E          E' 

~  R  +  r  ~'~  R'  (36) 

Substituting  either  of  these  values  for  /,  preferably  the  latter,  in 
eq.  (35),  we  have 

E-E'  =  E'^  (37) 


and  solving  for  r,  we  have 


r  =  R 


E  -  E' 
E' 


(38) 


Since  the  right-hand  member  of  this  equation  contains  a  ratio  of 
voltages,  it  is  not  necessary  to  know  actual  values,  relative 
values  being  sufficient;  hence,  any  device  giving  indications 


ELECTRICITY  AND  MAGNETISM 


proportional  to  the  voltage  may  be  used  in  place  of  the  volt- 
meter; for  example,  a  condenser  and  ballistic  galvanometer. 
46.  Condenser  and  Ballistic  Galvanometer  Method.  —  The 
basis  for  this  method  is  that  the  first  throw  of  the  galvanometer 
is  proportional  to  the  quantity  of  electricity  discharged  through 
it,  and  that  the  charge  of  a  condenser  is  proportional  to  the 
potential  difference  across  its  terminals;  that  is 

Q  =  CE 

where  C  is  the  capacity  of  the  condenser.  Accordingly,  if  the 
voltages  E  and  Ef  are  used  to  charge  the  condenser,  and  these 
charges  are  then  passed  through  the  ballistic  galvanometer,  the 
deflections  are  proportional  to  the  voltages;  that  is 


Q  = 


=  CE 


or 


E  =  k2d 
Substituting  in  (38),  we  have 

R(E  -  E') 


r  = 


E1 


d' 


(39) 
(40) 


47.  Experiment  4.     Internal  Resistance  of  a  Cell  by  Condenser 
Method. — Connect  the  apparatus  as  shown  in  Fig.   24,   where 

B  is  the  cell  to  be  tested,  C  a  con- 
denser, G  a  ballistic  galvanometer, 
K2  a  charge  and  discharge  key,  and 
R  a  known  variable  resistance.  First, 
with  KI  open,  press  down  K*  thus 
charging  the  condenser  to  the  total 
E.M.F.  of  the  cell,  and  discharge  by 
allowing  K%  to  rise,  obtaining  a  deflec- 
tion proportional  to  E.  Take  several 
readings  in  this  manner  and  average. 
Then,  having  set  R  at  a  suitable 
value,  close  K\,  charge  and  discharge 


VWWWWV 


FIG  24.— Condenaer  method  for     ag    abo          opening   K1    as    quickly  as 
internal  resistance  of  a  cell.  .  ,         ,      .    . 

possible  to  avoid  polarizing  the  cell. 

The  average  of  several  readings  taken  in  this  manner  measures 
E',  whence  r  may  be  computed.  It  is  well  first  to  practice 
operations  upon  a  cell  other  than  the  one  to  be  tested,  in  order 
to  become  expert  in  manipulating  the  keys  quickly  and  in 
their  proper  order.  In  carrying  out  this  experiment,  use  the 


MEASUREMENT  OF  RESISTANCE  53 

following  values  for  R:  10,  7,  5,  4,  3,  2,  1,  .5  and  .2  ohms.     The 
current  from  the  cell  is  given  by  — 


where  E  is  the  total  E.M.F.  To  obtain  E,  it  is  necessary  to 
determine  the  voltage  constant  of  the  condenser  and  galvano- 
meter system,  which  is,  in  realitv,  the  constant  of  eq.  39.  In 
other  words,  we  must  measure  the  voltage  required  to  give  unit 
deflection.  For  this  purpose  replace  B  by  a  standard  cell  and, 
with  KI  open,  charge  and  discharge  several  times.  Substitute  in 
eq.  (39)  and  solve  for  &2. 

Report.  —  1.  Compute  the  internal  resistance  for  each  differ- 
ent current  drawn  from  the  cell  and  plot  the  former  as  ordinates 
against  the  latter  as  abscissas. 

2.  How  do  you  account  for  the  fact  that  the  internal  resistance 
is  not  constant? 

48.  Battery   Test.  —  When   a    primary   battery   is   furnishing 
current,  it  polarizes;  that  is,  hydrogen,  which  is  one  of  the  products 
of  the  reactions  going  on  within,  collects  on  the  positive  plate. 
This,  together  with  other  causes,  diminishes  the  activity  of  the 
cell.     Indeed,  the  polarization  may  become  so  great  as  to  cause 
the  E.M.F.  to  fall  to  zero.     A  chemical,  called  the  depolarizer, 
is  introduced  to  remove  the  hydrogen,  or  to  prevent  its  being 
formed.     Cells  intended  for  open  circuit  work  contain  a  depolariz- 
ing agent  that  acts  very  slowly;  thus  they  polarize  rapidly  if  left 
on  closed  circuit,  but  recover  if  left  for  a  time  on  open  circuit. 
Cells    intended    for    closed    circuit  work   should   polarize  very 
little,    thus   th6    depolarizing   agent   should  act  quickly.     The 
deterioration  of  a  cell,  when  left  on  open  circuit,  due  to  local  action 
within  the  cell,  is  important,  but  can  best  be  found  by  actual  use, 
since  it  takes  too  long  to  test  this  in  the  laboratory.     We  might  also 
run  an  efficiency  test  by  working  the  cell  to  exhaustion;  but  this, 
too,  is  better  found  by  actual  use.     What  we  are  interested  in, 
however,  is  the  behavior  of  the  cell  when  run  on  a  closed  circuit 
for  a  given  time  as  the  value  of  a  cell  is  determined  by  the  rate 
of  its  polarization  and  recovery  as  well  as  by  its  E.M.F.  and 
internal  resistance. 

49.  Experiment  5.     Battery  Test.1  —  Study  in  this  experiment 
the  time  variation  of:  (a)  Total  E.M.F.  on  open  circuit,  (6)  the 
terminal  potential  difference  on  closed  circuit,  (c)  the  internal 

1  CARHART,  Primary  Batteries. 


54  ELECTRICITY  AND  MAGNETISM 

resistance,  (d)  the  current,  and  (e)  the  rate  of  recovery  from 
polarization.  The  set-up  is  the  same  as  in  Exp.  4  and  all 
quantities  are  to  be  measured  by  the  methods  there  outlined. 
The  difference  here  is  that  the  key  KI  is  left  closed  all  the  time 
except  for  an  instant  when  it  is  opened  to  charge  the  condenser 
for  measuring  the  total  E.M.F.  As  above,  obtain  the  readings 
for  the  E.M.F.  and  terminal  potential  difference  for  the  initial 
condition  of  the  cell.  Now  close  KI,  and  at  the  end  of  a  minute, 
charge  the  condenser  and  discharge  it  through  the  galvanometer, 
thus  obtaining  the  value  of  the  terminal  potential  difference  on 
closed  circuit  after  the  cell  has  been  delivering  current  for  one 
minute.  As  soon  as  this  is  done,  open  KI  for  an  instant  and 
charge  the  condenser;  then  set  Kz  on  " insulate"  and  again  close 
KI.  This  key,  K\,  must  be  opened  and  closed  quickly,  also 
these  two  readings,  i.e.,  for  total  E.M.F.  and  terminal  P.D., 
taken  as  nearly  simultaneously  as  possible.  As  soon  as  the 
galvanometer  comes  to  rest,  discharge  the  condenser  through  it, 
obtaining  a  measure  of  the  E.M.F.  on  open  circuit  after  the  cell 
had  been  furnishing  current  for  one  minute.  This  will  give  a 
measure  of  the  polarization.  Repeat  these  readings  every  minute 
for  five  minutes  and  then  every  five  minutes  for  twenty- 
five  minutes  more.  At  the  end  of  this  time,  open  KI  and 
measure  the  E.M.F.  of  the  cell  as  it  recovers  for  another  thirty 
minutes.  As  above,  take  readings  at  first  every  minute  and  then 
at  intervals  of  five  minutes.  Find  out  from  the  instructor  what 
resistance  to  use  for  R.  Read  Exp.  4  before  attempting  this  one. 
Practice  with  another  cell  as  there  suggested. 

Report. — 1.  Compute  the  total  E.M.F.  and  terminal  potential 
difference  in  volts,  and  internal  resistance  in  ohms. 

2.  Plot  on  one  sheet,  with  time  as  abscissas,  the  total  E.M.F., 
the  terminal  potential  difference,  the  internal  resistance,  and  the 
current  as  ordinates. 

3.  Plot  also  on  the  same  sheet  the  recovery  curve,  starting 
at  the  other  end  of  the  time  axis,  running  the  curve  backwards. 


CHAPTER  IV 


MEASUREMENT  OF  POTENTIAL  DIFFERENCE 

£< 

50.  Description  of  a  Potentiometer.1 — There  is,  perhaps,  no 
single  electrical  instrument  which  has  so  wide  a  field  of  usefulness 
and  which  gives,  at  the  same  time,  such  trustworthy  results  as 
the  potentiometer.  While  comparing  potentials  primarily,  it 
may,  with  proper  accessories,  be  adapted  to  compare  currents  and 
resistances  as  well,  and  is  so  easy  to  manipulate  as  to  be  an 
effective  instrument  even  in  the  hands  of  a  novice.  The  funda- 


M 


c 

•wwww 


-B 


N 


+  I- 

FIG.  25. — Simple  Potentiometer  circuit. 

mental  principle  of  the  potentiometer  may  be  illustrated  by  Fig. 
25,  where  MN  is  a  wire  of  uniform  resistance,  stretched  along  a 
scale  with  equal  divisions  and  supplied  with  current  from  a 
battery  B,  whose  E.M.F.  must  be  larger  than  those  to  be  com- 
pared. If  the  polarity  of  B  is  as  shown,  M  will  be  at  a  higher 
potential  than  N,  and  the  fall  of  potential  per  unit  length  will  be 
the  same  all  along  the  wire.  If  the  difference  of  potential  between 
M  and  N  is  known,  the  wire  may  be  regarded  as  a  potential  measur- 

1  LAWS,  Electrical  Measurements,  p.  271. 
Electrical  Meterman's  Handbook,  p.  208. 
KARAPETOFF,  Experimental  Engineering,  p.  74. 

55 


56  ELECTRICITY  AND  MAGNETISM 

ing  rod.  To  measure  an  unknown  E.M.F.,  such  as  the  battery  X 
of  the  figure,  an  auxiliary  circuit  MXL  is  provided,  containing  a 
galvanometer  and  key.  If  the  battery  X  were  temporarily  re- 
moved and  a  short  circuiting  wire  substituded  in  its  place,  a  por- 
tion of  the  current  from  B  would  flow  in  the  shunt  circuit  from  M 
to  L,  causing  a  deflection  of  the  galvanometer  in  a  particular 
direction .  If,  instead,  the  battery  B  were  removed,  X  would  cause 
a  current  to  flow  in  the  direction  XMLG,  giving  a  reverse  deflect- 
ion. If,  however,  both  batteries  are  included,  and  the  slider  L  is 
adjusted  until  the  /  R  drop  in  the  wire  due  to  the  current  from  B  is 
exactly  equal  to  the  E.M.F.  of  X,  no  current  will  flow  in  the  shunt, 
indicated  by  zero  deflection  of  the  galvanometer.  The  current  in 
the  circuit  BNM  is  then  just  the  same  as  though  the  shunt  were 
disconnected.  If  the  potential  drop  per  unit  length  of  the  slide 
wire  is  known,  X  may  be  directly  determined,  for  we  have 

X  =  pIZ,,  (1) 

where  p  is  the  resistance  per  unit  length,  and  li  the  length  required 
for  balance,  pi  may  be  determined  by  substituting  for  X  a  cell 
S  of  known  E.M.F.  and  balancing  as  before.  Let  I*  be  the  length 
required  for  this  balance.  Then 

S  =  pll*  (2) 

Whence 

.  P/  =  f      .       •        .  (3) 

and 

X  =  S12  (4) 

The  unknown  E.M.F.  is  thus  obtained  in  terms  of  S  by  a  direct 
comparison  of  the  lengths  li  and  12.  If  the  fall  of  potential  per 
unit  length  of  wire  were  some  decimal  fraction  of  a  volt,  the 
unknown  X  could  be  read  from  the  slide  wire  directly,  thus  avoid- 
ing the  calculation  indicated.  The  method  of  accomplishing 
this  may  be  illustrated  by  the  following  example:  Suppose  the 
slide  wire  MN  contains  200  divisions,  and  the  fall  of  potential 
between  M  an  AT  is  2  volts.  The  fall  of  potential  per  division 
is  then  .01  volt.  Let  the  standard  cell  have  an  E.M.F.  of 
1.0185  volts.  Set  the  slider  at  101.85  divisions,  include  S  in  the 
shunt  circuit,  and  obtain  a  balance,  not  by  moving  the  slider, 
but  by  varying  the  control  resistance  C,  thereby  changing  the 

current  /.     When    a    balance    has    been    secured,  pi  =    '  .  g- 


MEASUREMENT  OF  POTENTIAL 


57 


.01  volt  per  division.  The  potentiometer  is  now  standardized. 
Substitute X  for S  and  balance  by  moving!/,  leaving  C  unchanged. 
If  this  reading  should  be  145.63  divisions,  the  E.M.F.  of  the 
unknown  cell  would  be  1.4563  volts.  When  used  in  this  manner, 
the  instrument  is  said  to  be  a  "  Direct  Reading  Potentiometer. " 
To  carry  out  comparisons  with  great  accuracy,  a  very  long  wire, 
having  a  high  degree  of  uniformity,  is  obviously  necessary. 
Since  such  a  wire  is  difficult  to  obtain,  and  inconvenient  to  use,  it 
is  customary  to  substitute  for  it  two  resistances,  as  shown  in 
Fig.  26.  If  the  sum  of  R i  and  #2  is  kept  constant,  they  together 


M 


— W\MAA 


— VWWW 


+    I  — 
FIG.  26. — Potentiometer  constructed  from  resistance  boxes. 

are  equivalent  to  a  wire  of  fixed  length,  and  an  increase  in  RI 
accompanied  by  an  equal  decrease  in  R%  is  equivalent  to  moving 
the  slider  of  Fig.  25  to  the  right,  while  an  increase  in  R%  and  a 
decrease  in  R\  moves  it  to  the  left.  Balances  may  easily  be  ob- 
tained, the  conditions  for  which  are  the  same  as  outlined  above. 
For  example,  when  a  balance  has  been  obtained  with  -X"  in  circuit, 
we  have 

X  =  Rli.  (5) 

where  RI  is  the  resistance  required  for  balance  and  i,  the  current 
flowing  through  the  potentiometer,  i.e.,  through  CRiR%.  This 
current,  which  may  be  obtained  by  balancing  against  the  stand- 
ard cell  S,  is  given  by 

S  =  RSi  or  i  =      r  (6) 


58  ELECTRICITY  Afrb  MAGNETISM 

where  R'\  is  the  value  required  for  balance  in  this  case.     Whence, 


It  must  constantly  be  borne  in  mind  that  the  above  relations 
require  that  i  should  remain  constant  during  the  entire  process, 
which  will  be  true  only  when  the  sum  of  the  resistances,  R\+ 
R2  =  C  is  unchanged,  and  the  E.M.F.  of  B  is  constant.  This 
arrangement  may  be  made  "Direct  Reading"  if  i  is  a  known  dec- 
imal fraction  of  an  ampere.  This  may  be  accomplished  by  giving 
to  R'i  a  value  having  the  same  significant  figures  as  the  E.M.F. 
of  the  standard  cell,  and  balancing  by  varying  C.  For  example, 
suppose,  as  above,  S  —  1.0185  volts  and  the  boxes  used  have 
resistances  in  the  neighborhood  of  20,000  ohms.  If  #'1  =  10,185 
ohms,  when  a  balance  has  been  reached 
1.0185  1 


and  the  fall  of  potential  across  each  ohm  is  in  nnrt  of  a  volt. 

lUjUUU 

Replacing  now  SbyX  and  balancing  again,  leaving  C  undisturbed 
and  keeping  R\  +  Rz  constant, 

x  =  10^00  (8) 

51.  Standard  Potentiometers.  —  In  order  to  avoid  the  necessity 
of  providing  two  exactly  similar  boxes,  making  the  various  con- 
nections as  explained  above,  and  transferring  plugs  from  one  to 
the  other,  it  is  convenient  to  have  a  single  instrument,  including 
all  resistances,  switches,  keys,  etc.,  provided  with  binding  posts, 
to  which  the  various  E.M.F.  's  may  be  connected.  A  number  of 
such  potentiometers  are  on  the  market,  three  of  which  will  be 
described. 

1.  The  Leeds  and  Northrup  Potentiometer.  —  The  arrangement 
of  this  circuit,  which  is  the  simplest  of  those  to  be  studied,  is 
shown  diagrammatically  in  Fig.  27,  in  which  the  letters  corre- 
spond, as  far  as  possible,  to  those  used  in  Fig.  26.  The  potentio- 
meter circuit  proper,  BNMC,  consists  of  16  coils  of  5  ohms  each, 
and  a  long  slide  wire  NO,  also  of  5  ohms.  This  circuit,  in  normal 
operation,  carries  one  fiftieth  of  an  ampere,  giving  across  each 
coil  as  well  as  the  slide  wire,  one-tenth  of  a  volt  fall  of  potential. 
The  circuit,  containing  the  unknown  potential  is  included  between 
the  movable  contacts,  T  and  L.  The  box  RI  of  the  previous 


MEASUREMENT  OF  POTENTIAL 


59 


diagram  is  that  part  of  the  circuit  lying  between  T  and  L, 
while  Rz  consists  of  two  parts,  namely,  the  right-hand  portion  of 
the  slide  wire,  and  the  resistance  to  the  left  of  T.  The  slide 
wire,  shown  in  the  figure  by  a  single  turn,  in  reality  consists  of 
ten  turns  wound  on  a  marble  cylinder  and  is  about  17  feet  in 
length.  The  fall  of  potential  across  each  turn  is  thus  .01  volt, 
and,  as  the  dial  circle  is  dividedjnto  200  parts,  the  instrument 


ST'D  CEIL  UNKNOWN 

FIG.  27. — Diagram  of  Leeds  and  Northrup  potentiometer. 

reads  to  directly  .00005  volt.  By  moving  the  slider  L  and  the  con- 
tact T,  the  difference  of  potential  may  be  varied  by  infinitesimal 
changes  from  0  to  1.6  volts. 

The  range  of  the  instrument,  for  small  electromotive  forces, 
such  as  those  furnished  by  thermo-couples,  is,  increased  tenfold 
by  means  of  a  shunt.  As  seen  from  the  diagram,  when  the  plug  P 
is  inserted  in  the  hole  marked  .1,  the  shunt  S,  which  contains  a 
resistance  of  one-ninth  that  of  the  potentiometer  proper,  is  con- 
nected across  the  entire  circuit,  so  that  only  one-tenth  of  the 
normal  current  flows  through  the  potentiometer  proper.  In 
order  that  the  current  from  the  battery  B  may  remain  unchanged  a 
resistance  K  is  automatically  included,  thus  keeping  the  resis- 
tance of  the  entire  circuit  the  same.  A  ready  means  of  standard- 
izing the  potentiometer  current  is  furnished  by  the  extra  dial 
DE,  containing  19  coils  of  such  a  resistance,  that,  with  the 


60  ELECTRICITY  AND  MAGNETISM 

normal  current  flowing,  the  fall  of  potential  across  each  is  .0001 
volt.  From  the  .6  post  of  the  tenth  volt  dial,  a  permanent  lead 
is  brought  out,  and  connected,  when  the  selecting  switches  are 
thrown  toward  the  left,  through  the  galvanometer  and  standard 
cell  to  the  contact  F.  The  fall  of  potential  from  the  .6  plug  to 
M  is  1  volt;  from  M  through  the  resistance  to  E  is  .018  volt,  and 
from  E  to  F  as  many  ten-thousandths  of  a  volt  additional  as  may 
be  required  to  equal  the  E.M.F.  of  any  normal  Weston  cell 
within  the  ordinary  range  of  temperature  (1.0180-1.0204  volts). 


i 


FIG.  28. — Leeds  and  Northrup  potentiometer. 

It  is  thus  possible  to  check  the  potentiometer  current  without 
re-setting  the  instrument.  The  operation  then  is  as  follows: 
Set  the  standard  cell  dial  to  correspond  to  the  E.M.F.  of  the  cell, 
corrected  for  temperature.  Move  the  selecting  switch  to  the 
left,  set  P  in  the  hole  marked  1,  and  vary  the  control  resistance 
C  (usually  mounted  at  the  right-hand  end  of  the  instrument) 
until  a  balance  is  obtained.  One-fiftieth  of  an  ampere,  the  nor- 
mal current  is  now  flowing.  Move  the  selecting  switch  to  the 
right,  thus  including  the  unknown  E.M.F.,  and  vary  T  and  L 
until  the  balance  is  once  more  obtained,  when  the  unknown  may 
be  read  directly.  If  it  is  less  than  .15  volt,  set  P  in  the  .1  hole, 
and  balance  again,  when  the  reading  of  the  instrument  must  be 
divided  by  10.  The  complete  instrument  is  shown  in  Fig.  28. 

2.  The  Wolff  Potentiometer. — The  fundamental  principle  of 
this  instrument  is  shown  by  the  simplified  connections  of  Fig. 
29.  The  result  which  must  be  secured  by  any  arrangement  is 


MEASUREMENT  OF  POTENTIAL 


61 


that  the  resistance  of  the  potentiometer  circuit  proper,  namely, 
MN,  must  be  kept  constant,  while  the  resistance  across  which 
the  auxiliary  circuit  FL  is  connected,  must  be  continuously 
variable.  By  moving  F  and  L,  changes  of  1,000  and  100  ohms, 
respectively,  are  obtained,  without  changing  the  total  resistance 
as  is  at  once  obvious.  The  ^resistance  coils  connected  by  the 
double  sliders  are  sets  with  units  of  10,  1,  and  .1  ohms  respec- 
tively. These  double  sliders  are  mounted  so  as  to  move  together, 
but  are  insulated  from  each  other  and  connected  in  circuit  as 


vVWWWVWWVW- 
vvwwwvp 


FIG.  29. — Principle  of  Wolff  potentiometer. 

shown  in  the  diagram.  If  the  pair  at  the  left  is  moved  one  divi- 
sion to  the  right,  it  is  evident  that  the  resistance  between  N  and 
L  is  increased  by  10  ohms,  while  that  between  F  and  L  is  decreased 
by  the  same  amount,  thus  leaving  the  resistance  between  M 
and  N  unchanged.  In  the  same  way,  the  middle  pair  of  sliders 
produces  changes  of  1  ohm  each  between  F  and  L  leaving  MN 
unchanged,  while  the  right-hand  pair  produces  changes  of  .1 
ohm  each.  Shifting  any  one  of  these  sliders,  therefore,  is  equi- 
valent to  moving  the  slider  L  of  Fig.  25  by  definite  amounts. 

The  actual  wiring  of  the  instrument,  mounting  of  the  sliding 
contacts,  connections  to  accessories,  switches,  etc.,  are  shown  in 
Fig.  30.  The  control  resistance  K  is  not  included  in  the  instru- 
ment. Any  ordinary  resistance  box  capable  of  small  variations 
will  serve  for  this  purpose.  The  total  resistance  of  the  instru- 
ment, as  sketched,  is  19,000  ohms.  When  carrying  the  normal 
current  of  one  ten-thousandth  of  an  ampere,  the  difference  of 


62 


ELECTRICITY  AND  MAGNETISM 


potential  across  consecutive  posts  of  the  first  dial  is  one-tenth 
of  a  volt;  of  the  second  dial,  one-hundredth  of  a  volt;  and  of  the 
last  dial,  one  hundred-thousandth  of  a  volt,  while  the  maximum 
voltage  directly  measurable  is  one  and  nine-tenths  volts. 

In  using  this  instrument,  first  obtain  the  E.M.F.  of  the  stand- 
ard cell,  corrected  for  temperature,  and  set  the  small  middle  dial 
of  the  upper  row  at  this  value.  Set  the  switch  in  the  upper  left- 
hand  corner  at  NN,  and  the  one  in  the  right-hand  corner,  which 
includes  a  high  resistance  in  the  galvanometer  circuit,  at  its 
largest  value.  Obtain  a  balance  by  varying  the  control  resis- 


FIG.  30. — Wiring  diagram  for  Wolff  potentiometer. 

tance,  cutting  out  the  galvanometer  resistance  as  a  balance  is 
approached.  This  operation  standardizes  the  current  at  one 
ten-thousandth  of  an  ampere.  Now  switch  to  XX  and  balance 
by  setting  the  large  dials,  when  the  unknown  may  be  read  off 
directly.  In  checking  the  potentiometer  current,  which  must 
frequently  be  done,  it  is  not  necessary  to  change  the  dials 
from  their  positions  when  balanced  on  the  unknown  E.M.F. 

3.  The  Tinsley  Potentiometer. — The  working  diagram  for  this 
instrument,  which  is  unique  in  that  it  employs  an  electrical 
vernier,  is  shown  in  Fig.  31.  Seventeen  coils,  with  a  resistance 
of  5  ohms  each,  connected  in  series  with  a  short  slide  wire  of  .5 
ohm,  form  the  potentiometer  circuit  proper  MN,  while  the  auxil- 
iary circuit  is  FGL.  Attached  to  a  movable  arm  are  two  sliding 
contacts,  so  spaced  that  they  always  rest  upon  two  alternate 
posts,  leaving  one  post  between  them  as  indicated.  This  pair  of 
contacts  is  connected  to  a  second  series  of  10  coils  of  1  ohm  each. 


MEASUREMENT  OF  POTENTIAL  63 

When  the  normal  current  of  one-fiftieth  of  an  ampere  is  flowing 
through  MNj  the  fall  of  potential  between  adjacent  posts  is  .1 
volt.  However,  the  fall  of  potential  between  the  posts  connected 
by  the  pair  of  contacts  to  the  second  series  of  coils  is  also  .1  volt, 
since  the  two  coils  of  the  main  circuit  are  now  shunted  by  a  resis- 
tance equal  to  their  own,  giving  a  resultant  resistance  between  the 
contacts  equal  to  that  of  a  single  main  circuit  coil.  Between 
adjacent  posts  of  the  second  series  there  is  accordingly  .01  volt 
fall  of  potential,  and  across  the  slide  wire  there  is  also  .01  volt 
potential  difference.  This  instrument,  like  the  Leeds  and  North- 
rup,  is  provided  with  separate  connections  for  the  standard  cell, 


RMEOSTAT-J^ 



f*|                           .1  VOLT 

f       ¥ 

L                 f 

I              

1  l-Ma 

|T     «|S]4|3|2|.|0|  1 

F 

f      .01  VM.T 

—       ,         ^ 

POT*  > — < 

FIG.  31. — Principle  of  Tinsley  potentiometer. 

so  that  it  is  not  necessary  to  re-set  all  of  the  sliders  when  checking 
the  current  through  the  potentiometer  circuit  proper.  A  stand- 
ard cell  lead  is  permanently  attached  to  post  number  7.  Across 
the  ten  coils  between  it  and  post  17  there  is,  accordingly,  1  volt 
potential  difference,  and  in  series  with  the  main  circuit  is  another 
coil  shown  at  the  left  of  17,  of  such  a  value  that,  with  the  normal 
current  flowing,  the  fall  of  potential  across  it  is  .0183  volts,  and  to 
the  other  side  of  this,  the  second  standard  cell  terminal  is  attached. 
Unlike  the  Leeds  and  Northrup  instrument,  this  coil  cannot  be 
varied  to  compensate  for  variations  in  the  E.M.F.  of  the  standard 
cell  due  to  temperature  changes,  but  the  value  1.0183  volts  is 
sufficiently  accurate  for  ordinary  purposes. 

The  wiring  connections,  switches,  etc.,  are  shown  in  Fig.  32. 
The  control  rheostat  is  included  in  the  instrument,  and  consists 
of  the  dial  in  the  right-hand  corner  and  the  slide  wire  immediately 
above  it.  By  moving  the  plug  in  the  upper  left-hand  corner  to 
the  hole  marked  X  by  .1,  the  instrument  is  shunted  by  a  resis- 
tance of  such  a  value  that  all  readings  should  be  divided  by  ten,  a 


64 


ELECTRICITY  AND  MAGNETISM 


feature  of  great  importance  in  thermo-couple  work.  In  using 
the  instrument,  set  the  shunt  plug  in  the  hole  marked  X  by  1, 
and  the  two-point  switch  below  the  middle  dial  on  SC.  The 
first  dial  must  be  set  so  as  to  shunt  none  of  the  coils  above  the 
seventh,  otherwise,  the  resistance  over  which  the  standard  cell  is 
to  be  balanced  will  be  reduced  effectively  by  one  coil.  A  good 
rule  is  to  set  this  dial  always  at  zero  when  balancing  on  the  stand- 
ard cell.  Obtain  a  balance  by  changing  first  the  rheostat  and 
then  the  slider  above  it,  which  is  provided  with  a  slow  motion 
screw  for  the  final  setting.  Use  in  this  connection  the  key  mark 


POT.  KEY  «S.C.  KEY 

FIG.  32. — Wiring  diagram  of  Tinsley  potentiometer. 

SC.  The  current  is  now  exactly  one-fiftieth  of  an  ampere  and  the 
instrument  is  standardized.  To  measure  the  unknown,  simply 
move  the  two-point  switch  to  Potl.,  and  balance  by  setting  the 
two  dials  and  the  lower  slide  wire.  If  the  unknown  is  less  than  .2 
volt,  use  the  shunt  as  explained  above,  dividing  the  reading  by  10. 
In  carrying  out  any  measurement,  the  current  through  the 
instrument  should  be  checked  frequently. 

52.  The  Weston  Standard  Cell. — While  the  legal  definition 
of  electromotive  force  is  given  in  terms  of  the  standard  current 
and  resistance  by  means  of  Ohm's  law,  nevertheless,  in  actual 
practice,  the  volt  is  specified  in  terms  of  the  standard  cell.  After 
many  years  of  investigation,  the  Weston  standard  cell  has  been 
perfected  to  such  an  extent  that  persons  in  different  parts  of  the 


MEASUREMENT  OF  POTENTIAL 


65 


world,  may,  by  following  definite  specifications,  construct  cells 
of  this  type  and  be  sure  of  securing  E.M.F.'s  which  agree  within 
less  than  1  part  in  10,000.  This  cell  is  usually  set  up  in  an  air-tight 
H -shaped  vessel,  as  shown  in  Fig.  33,  with  platinum  wires  sealed 
through  the  bottoms  for  connection  with  the  electrodes.  The 
positive  electrode  consists  of  pure  mercury  while  the  negative  is 
an  amalgam  of  cadmium  and  mercury.  These  are  placed  in  the 
bottoms  of  the  tubes,  ^and  a  solution 
of  CdSC>4  with  a  few  extra  crystals  to 
insure  saturation,  forms  the  electro- 
lyte between  them.  To  protect  the 
mercury  against  contamination  by 
the  CdSO4  and  at  the  same  time 
prevent  polarization,  a  thick  paste, 
consisting  mainly  of  mercurous  sul- 
phate ,  is  placed  over  the  mercury.  As 
the  cell  operates,  the  cadmium  ions 
from  the  CdSO4  solution  displace 
some  of  the  ions  from  the  mercurous 
sulphate  paste  and  mercury  is  de-  FIG.  33.— Weston  standard  cell, 
posited  upon  the  mercury  electrode. 

One  of  the  advantages  of  this  cell  over  former  types  is  that  its 
electromotive  force  changes  but  very  little  with  the  temperature. 
The  electromotive  force  of  a  cell  which  has  been  set  up  with  care  is 
given,  with  accuracy  sufficient  for  most  purposes,  by  the  equation 


Et  =  #20  -  0.0000406  (t  -  20°  C.). 


(9) 


That  is,  the  E.M.F.  decreases  0.0000406  volt  for  each  degree  the 
temperature  is  raised  above  20°  C.,  and  increases  by  the 
same  amount  for  each  degree  below  20°  C.  This  quantity  is 
called  the  temperature  coefficient.  Since  standard  cells  are  never 
used  as  a  source  of  current,  but  merely  for  balancing  potentials  or 
charging  condensers,  they  are  made  of.  small  size.  Those  fur- 
nished in  the  laboratory  are  mounted  in  a  brass  tube,  with  a  hard 
rubber  top  provided  with  binding  posts  and  a  hole  through  which 
to  insert  a  thermometer.  The  E.M.F.  of  the  individual  cells 
is  usually  given  at  20°  C.,  from  which  the  E.M.F.  at  the  temper- 
ature at  which  they  are  used  may  be  computed  by  means  of  the 
formula  given  above.  When  used  in  a  potentiometer  circuit, 
a  high  resistance  should  be  included  and  gradually  cut  out  as  a 
balance  is  approached. 


66  ELECTRICITY  AND  MAGNETISM 

53.  Experiment  6.  Comparison  of  Cells  by  the  Potentio- 
meter. A.  Simple  Potentiometer. — Connect  the  apparatus,  as 
shown  in  Fig.  26,  Art.  50,  omitting  the  control  resistance  C, 
and  using  for  Ri  and  Rz  two  exactly  similar  boxes.  B  should  be 
a  cell  of  constant  E.M.F.,  preferably  a  portable  storage  battery. 
Obtain  from  the  instructor  a  standard  and  several  unknown  cells 
whose  E.M.F.'s  are  to  be  determined.  The  high  resistance 
marked  H.R.  need  not  be  known  accurately,  since  its  purpose  is 
merely  to  protect  the  galvanometer  and  standard  cell  from 
excessive  currents  when  the  potentiometer  is  far  from  balance. 
It  is  well  to  start  this  at  about  10,000  ohms,  gradually  reducing 
it  as  a  balance  is  approached.  Be  sure  that  the  double  pole 
double  throw  switch  for  connecting  S  and  X  in  circuit  is  not 
provided  with  cross  wires,  as  they  would  short  circuit  the  cells. 
To  keep  RI  -{-  Rz  constant,  as  required  in  the  theory,  start  with 
all  the  plugs  out  of  R\  and  all  in  #2,  and  obtain  a  balance  by 
transferring  them  from  their  places  in  one  box  to  the  correspond- 
ing holes  in  the  other.  Ri  +  Rz  will  then  always  remain  equal  to 
the  total  resistance  of  one  box.  To  test  whether  the  polarity  of 
the  cells  is  properly  arranged  in  the  two  circuits,  first  rock  the 
double  pole  double  throw  switch  on  X,  break  the  potentiometer 
circuit  at  B}  tap  the  key  K  lightly,  and  note  the  direction  of 
swing  of  the  galvanometer.  Now  close  again  the  circuit  at  B, 
remove  the  wires  from  the  middle  posts  of  the  double  pole  double 
throw  switch,  and  join  them.  The  galvanometer  should  swing 
in  the  opposite  direction  on  tapping  the  key.  First,  secure  a 
balance  on  X',  then  rock  the  switch  over  and  balance  on  S, 
afterwards  checking  your  balance  on  X,  to  make  sure  that  the 
potentiometer  current  has  not  changed  during  the  process. 
Reverse  the  connections  at  B,  also  on  the  auxiliary  circuit,  and 
proceed  as  before,  taking  the  average  of  the  two  results  thus 
obtained.  This  is  necessary  to  eliminate  errors  due  to  spurious 
contact  and  thermal  E.M.F.'s  within  the  potentiometer. 

B.  Direct  Reading  Potentiometer. — Include  in  the  potentio- 
meter circuit  the  control  resistance  C,  as  shown  in  Fig.  26. 
Determine  the  temperature  of  the  standard  cell  and  its  E.M.F. 
corrected  for  this  temperature.  Set  Ri  to  have  the  same  significant 
figures  as  this  E.M.F.,  using  the  largest  multiple  possible,  and 
put  Rz  equal  to  the  difference  between  the  total  capacity  of 
one  box  and  Ri.  Switch  $  into  the  shunt  circuit  and  balance  by 
varying  C.  Then  rock  over  on  to  X  and,  leaving  C  fixed,  balance 


MEASUREMENT  OF  POTENTIAL 


67 


by  plugging  back  and  forth  between  RI  and  R%,  keeping  their 
sum  constant.  The  reading  of  RI,  when  properly  pointed  off, 
gives  X  directly.  After  each  balance  on  X,  the  setting  on  the 
standard  cell  should  be  checked  and  C  changed,  if  the  current 
has  not  remained  constant,  which,  of  course,  necessitates  a  new 
balance  on  X.  Now  reverse  tterminals  as  in  Part  A,  and  repeat, 
taking  the  average  of  the  two  results. 

Report. — 1.  Give    values    of    E.M.F.  for  all  cells  compared, 
and  where  temperature  corrections  are  known,  reduce  to  20°  C. 

2.  Suppose  a  balance  has  been  obtained  without  H.R.   in 
circuit.     Now  include  H.R.     How  will  the  balance  point  be 
affected?     Why? 

3.  What  is  the  maximum  E.M.F.  that  may  be  measured 
by  the  direct  reading  potentiometer,  as  you  have  used  it  in  this 
experiment? 

54.  The  Volt  Box. — In  standard  potentiometers,  operated  on 
normal  current,  the  maximum  difference  of  potential  which  may 


FIG.  34.— Volt  box. 

be  measured  directly  never  exceeds  two  volts  and  is  usually  even 
less.  When  it  is  desired  to  measure  voltages  in  excess  of  this 
value,  some  means  must  be  provided  for  accurately  dividing  the 
unknown  voltage  into  definite  fractions  of  the  total,  small  enough 
to  be  measured  by  the  potentiometer  available.  This  may  be 


68 


ELECTRICITY  AND  MAGNETISM 


accomplished  by  means  of  the  "volt  box."  This  consists  of  an 
accurately  adjusted  resistance  box,  of  large  range,  in  which  the 
blocks  to  which  the  coils  are  attached  are  provided  with  sockets 
for  receiving  traveling  plugs.  The  voltage  to  be  divided  is 
impressed  across  the  terminals  and  the  fraction  to  be  measured 
is  obtained  across  the  traveling  plugs,  which  may  be  set  at  any 
points  desired.  By  Ohm's  law,  the  voltage  across  the  traveling 
plugs  is  such  a  fraction  of  the  total  voltage  as  the  resistance 
between  the  traveling  plugs  is  of  the  total  resistance.  If  the 
resistance  of  the  volt  box  is  10,000  ohms,  the  drop  across  1,000 
ohms  is  one-tenth  of  the  total;  that  across  100  ohms,  one-hun- 
dredth of  the  total,  and  so  on.  It  is  simpler  to  use  decimal  ratios 
wherever  practicable.  Special  boxes  are  made  in  which  these 
ratios  are  obtained  by  setting  a  dial  switch  or  a  single  plug  as 
shown  in  Fig.  34. 


'V\WVVXWWM\'VVWVWVV 


^VWvWWWWWVAAAAAAAAr 


FIG.  35. — Connections  for  standardizing  a  volt  meter. 

65.  Experiment  7.  Calibration  of  a  Voltmeter  by  Potentio- 
meter and  Volt  Box.1 — The  method,  in  brief,  consists  in  impress- 
ing across  the  terminals  of  the  voltmeter  various  voltages  and 
measuring  these  voltages  by  means  of  a  potentiometer  provided 
with  a  volt  box.  The  connections  for  this  purpose  are  shown  in 
Fig.  35.  VM  is  the  voltmeter  to  be  calibrated;  LO  the  volt 
box;  RS  a  high  resistance  rheostat  with  a  sliding  contact  for 

1  JANSKY,  Electrical  Meters,  chap.  V. 
KARAPETOFF,  Experimental  Engineering,  vol.  I,  pp.  51-55. 


MEASUREMENT  OF  POTENTIAL  69 

voltage  regulation,  and  B  a  storage  battery.  P  and  T  are  the 
terminals  from  the  traveling  plugs  of  the  volt  box  which  are  to  be 
attached  to  the  potentiometer.  The  voltage  of  B  should  be 
sufficient  to  give  full  scale  deflection  of  the  instrument.  Use 
any  one  of  the  potentiometers  described  above,  following  the 
directions  given  for  each  instrument.  After  the  connections 
with  the  potentiometer  have  been  properly  made  and  its  current 
adjusted  by  balancing-  againsf'  the  standard  cell,  throw  the 
selecting  switch  to  the  point  marked  "  unknown."  Set  the 
slider  of  the  rheostat  RS  so  that  the  voltmeter  indicates  about 
one-tenth  full  scale  deflection,  and  choose  the  largest  decimal 
ratio  of  the  volt  box  giving  a  voltage  within  the  range  of  the 
potentiometer.  Measure  this  voltage  with  the  potentiometer. 
In  a  similar  manner  check  the  voltmeter  at  8  or  10  points  dis- 
tributed uniformly  across  the  scale.  Test  the  constancy  of  the 
potentiometer  current  frequently  by  re-balancing  against  the 
standard  cell.  Record  voltmeter  readings,  potentiometer  set- 
tings, and  volt  box  ratios.  Note  carefully  the  zero  reading  of 
the  voltmeter  before  beginning  the  test  and  again  at  the  end, 
after  it  has  been  deflected  fqr  some  time,  to  see  if  the  springs 
show  any  elastic  fatigue.  With  about  two-thirds  full  scale 
deflection,  place  the  instrument  in  a  vertical  position  to  test  the 
accuracy  with  which  the  moving  system  is  balanced.  Bring 
another  instrument  near  this  one,  and  see  if  there  is  any  effect 
from  external  magnetic  fields.  Tap  the  instrument  gently  with 
the  finger  to  see  if  the  bearing  friction  is  large.  Does  the  pointer 
swing  past  its  final  position  when  a  voltage  is  suddenly  thrown 
on? 

Report. — 1.  Obtain  the  differences  between  the  readings  of 
the  instrument  and  true  voltages,  and  plot  these  corrections  as 
ordinates  against  readings  of  the  instrument  as  abscissas.  Draw 
in  straight  lines  connecting  these  points. 

2.  State   your   findings   regarding  the   imperfections   of   the 
instrument. 

3.  Would  it  indicate  on  alternating  voltages? 


CHAPTER  V 
MEASUREMENT  OF  CURRENT 

56.  Kelvin's  Balance. — This  is  an  instrument  for  the  measure- 
ment of  current  in  which  use  is  made,  not  of  the  action  between 
the  magnetic  field  of  a  current  and  a  permanent  magnet,  as  in 
the  case  of  galvanometers  and  ammeters,  but  of  the  action 
between  the  fields  of  two  currents.  It  consists  of  six  flat  coils 
placed  horizontally,  four  of  which  are  fixed  while  the  other  two, 
mounted  at  the  ends  of  a  beam  pivoted  at  the  middle,  are  movable. 
The  general  arrangement  is  shown  in  Fig.  36.  The  current  to  be 
measured  passes  through  all  six  coils  in  series,  flowing  in  each  in 


FIG.  36.- — Arrangement  of  coils  in  Kelvin's  balance. 

such  a  direction  that  A  and  C  both  urge  E  downward,  while  B  and 
D  urge  F  upward.  The  force  of  attraction  or  repulsion  between 
two  coils  is  proportional  to  the  current  in  each.  Accord- 
ingly, when  the  coils  are  connected  in  series  the  force  between 
them  is  proportional  to  the  square  of  the  current.  Thus,  the 
electrodynamic  action  between  the  fixed  and  the  movable  coils 
is  such  as  to  produce  a  torque  on  the  movable  ones  in  the  counter 
clockwise  direction  proportional  to  the  square  of  the  current. 
This  torque  is  counterbalanced  by  a  weight  which  slides  along  a 
graduated  beam  attached  to  the  moving  system.  An  index  at 
each  end  shows  when  a  balance  has  been  reached.  Since  the 

70 


MEASUREMENT  OF  CURRENT  71 

torque  due  to  the  current  is  proportional  to  the  square  of  the 
current,  and  that  due  to  the  weight  is  proportional  to  the  weight 
and  the  length  of  the  lever  arm,  we  have,  as  the  condition  for 
equilibrium, 

KP  =  WL  (1) 

where  W  is  the  weight  of  the  glider,  L  its  distance  from  the  zero 
position,  and  K  a  constant  depending  upon  the  construction  of 
the  instrument.  Solving 

-  I2  =  \  L  (2) 

or 

7  =  const.  VL  (3) 

The  constant  is  generally  so  given  that  one  must  use  the 
doubled  square  root  of  the  length  L,  and,  to  facilitate  observa- 
tions, tables  of  these  quantities  have  been  prepared.  For  rough 
work,  however,  a  fixed  scale  is  mounted  directly  behind  and  a 
little  above  the  movable  one,  from  which  the  doubled  square 
root  may,  with  fair  approximation,  be  read  directly.  Since  the 
constant  depends  upon  the  weight  of  the  slider,  a  means  is 
afforded  for  changing  the  range.  Four  weights  are  usually 
supplied  for  which  the  constants  are  0.025,  0.05,  0.1,  and  0.2, 
giving  ranges  of  1.25,  2.5,  5,  and  10  amperes,  respectively, 
since  the  movable  scale  has  625  divisions,  giving  a  doubled  square 
root  of  50. 

As  with  an  ordinary  balance,  the  beam  must  be  in  equilibrium 
for  no  load,  that  is,  no  current  flowing  through  the  coils.  If  the 
index  at  the  end  does  not  read  zero,  equilibrium  may  be  obtained 
by  moving  a  small  metal  flag  attached  to  the  moving  system  so 
as  to  throw  more  of  its  weight  to  one  side  or  the  other,  as  is 
required.  A  special  device  mounted  on  the  base  and  operated 
by  a  handle  below  the  case  is  provided  for  this  purpose.  The 
movable  system  is  carried  by  flexible  ligaments  made  up  of  a 
number  of  fine  phosphor-bronze  ribbons  placed  side  by  side. 
As  these  are  delicate  and  easily  broken,  an  arrestment  is  provided 
which  is  operated  by  a  milled  head  at  the  bottom  of  the  case. 
Weights  should  never  be  changed  without  first  raising  the 
arrestment.  Since  the  balance  must  be  in  equilibrium  for  zero 
current,  no  matter  which  weight  is  used,  there  must  be  a  separate 
counterpoise  for  each.  These  consist  of  brass  cylinders,  provided 
with  a  pin,  which  are  placed  in  a  small  horizontal  trough  at  the 
right-hand  end  of  the  moving  system,  with  one  end  of  the  pin 


72 


ELECTRICITY  AND  MAGNETISM 


passing  through  the  hole  in  the  bottom  of  the  trough.     Since  the 
direction  of  the  torque  is  independent  of  the  direction  of  the 


FIG.  37. — Kelvin's  balance. 

current,  the  instrument  may  be  used  either  on  direct  or  alternat- 
ing currents,  indicating  in  the  latter  case,  root  mean  square 
values.  Figure  37  shows  the  usual  laboratory 
form  of  the  Kelvin's  balance. 

57.  The  Siemens  Electrodynamometer. — 
This  is  another  current  measuring  instru- 
ment working  on  the  principle  of  the  electro- 
dynamic  action  between  two  coils  carrying 
currents.  The  coils  are  rectangular  in  form 
and  placed  perpendicular  to  one  another,  as 
shown  in  Fig.  38.  The  movable  coil,  CF, 
which  is  placed  outside  the  fixed  coil  AB,  is 
carried  by  a  fine  point  resting  in  a  jewel  and 
the  current  is  led  to  and  from  it  by  wires 
dipping  into  mercury  cups  at  a  and  b,  situated 
one  above  the  other  in  the  axis  of  rotation. 
One  end  of  a  helical  spring  S  is  attached  to 
the  moving  coil,  while  the  other  is  fastened 
to  a  milled  head  D  carrying  an  index  read 
from  a  fixed  circular  scale.  When  a  current 
flows  through  the  two  coils  in  series,  the 
movable  one  tends  to  set  itself  parallel  to  the  fixed,  but  is 
brought  back  to  its  zero  position  by  turning  the  head  D,  thus 


FIG 


—  A  r- 


38 

rangement  of  coils 
in  Siemens  electro- 
dynamometer. 


MEASUREMENT  OF  CURRENT 


73 


twisting  the  spring.  The  torque  due  to  the  current  is  propor- 
tional to  the  square  of  the  current  since  the  coils  are  in  series, 
while  that  due  to  the  spring,  by  Hooke's  Law,  is  proportional  to 
the  angle  through  which  it  is  twisted.  Accordingly,  we  have,  as 
the  condition  for  equilibrium, 


P  =*=  Az<f> 


or 


where  A  is  a  constant  depending  upon  the  size  of  the  coils,  number 
of  turns,  stiffness  of  spring,  etc,  and  <f>,  the  angle  through  which  the 
spring  is  twisted.  The  range  of  the  instrument  is  changed  by 
varying  the  number  of  turns  in 
one  of  the  coils.  The  instru- 
ment usually  has  two  fixed  coils 
with  separate  binding  posts  on 
the  base.  Since  the  magnetic 
field  of  these  coils  is  small,  that 
of  the  earth  is  appreciable  in 
comparison  and  may  introduce 
an  error.  For  example,  if  the 
earth's  field  is  in  the  same  direc- 
tion as  that  of  the  fixed  coil,  the 
instrument  will  read  too  high, 
while  if  the  earth's  field  is  oppo- 
site, it  will  read  too  low.  This 
error  may  be  eliminated  by  re- 
versing the  currents  and  taking 
the  average.  Since  the  direction 

of  rotation  of  the  movable  coil  is  independent  of  the  direction  of 
the  current,  the  instrument  will  indicate  on  alternating  currents 
as  well  as  direct,  giving  in  the  latter  case,  root  mean  square 
values.  It  may  accordingly  be  calibrated  on  direct  and  used 
on  either  direct  or  alternating  currents.  Figure  39  shows  the 
usual  form  of  Siemens  electrodynamometer. 

58.  Experiment  8.  Calibration  of  an  Electrodynamometer.1 — 
In  this  experiment,  an  electrodynamometer  is  to  be  calibrated 
in  terms  of  a  Kelvin  balance,  which  is  taken  as  the  standard 
instrument.  Connect  the  instruments  in  series  on  a  20-volt 

1  JANSKY,  Electrical  Meters,  chap.  VIII. 
CARHART  and  PATTERSON,  Electrical  Measurements,  chap.  III. 


FIG.  39. — Siemen's  electrodyna- 
mometer. 


74  ELECTRICITY  AND  MAGNETISM 

storage  battery,  including  a  variable  rheostat  and  an  ammeter 
to  observe  roughly  the  currents  used.  Both  instruments  must 
first  be  leveled  and  adjusted  for  zero  on  no  current.  Begin  with 
the  lowest  range  of  the  Kelvin  balance.  For  this  use  the  carriage 
alone  and  the  smallest  counter  weight.  When  the  limit  of  this 
range  has  been  reached,  raise  the  arrestment,  open  the  case,  and 
push  the  carriage  moving  mechanism  a  little  to  one  side  bringing 
it  forward  enough  for  clearance.  Place  the  first  additional  weight 
upon  the  carriage,  and  the  second  counter-poise  in  the  trough. 
Whenever  new  weights  are  put  in  position,  the  zero  must  be 
rechecked.  Measure  in  this  way  the  currents  for  ten  points  on 
the  electrodynamometer,  taking  them  a  little  closer  at  the 
lower  end  of  the  scale.  Record  electrodynamometer  readings, 
Kelvin  balance  readings,  and  number  of  counterpoise. 

Report. — 1.  Compute  the  current  for  each  setting  of  the 
instrument,  also  the  constant  A  in  equation  (5). 

2.  Plot  current  as  ordinates  and  settings  as  abscissas.     What 
is  the  shape  of  this  curve? 

3.  What    is    meant    by  the  root  mean  square  value  of  an 
alternating  current? 

4.  Name  some  other  electrical  instruments  operating  on  the 
principle  of  the  electrodynamometer. 

59.  Ammeters  and  Voltmeters.1 — An  ammeter,  as  the  name 
implies  is  an  instrument  for  measuring  the  current  flowing  in  a 
circuit;  while  a  voltmeter,  measures  the  difference  of  potential  or 


Line  [V.M.J  Load 


FIG.  40. — Connections  for  ammeter  and  voltmeter. 

electrical  pressure  existing  between  two  points  Since  the  former 
indicates,  at  any  instant,  the  rate  of  flow  of  electricity  through  a 
conductor,  it  must  be  placed  in  series  with  the  circuit,  so  as  to  be 
traversed  by  the  entire  current;  while  the  latter,  being  a  pressure 
gauge,  is  connected  in  parallel  with  the  circuit,  and  carries  a  very 
small  current,  which  in  general  may  be  neglected.  The  regular 
method  of  connecting  these  instruments  is  shown  in  Fig.  40. 

1  JANSKY,  Electrical  Meters,  chap.  III. 

KARAPETOFF,  Experimental  Electrical  Engineering,  vol.  I,  chap.  II. 
Electrical  Meterman's  Handbook,  chap.  V. 


MEASUREMENT  OF  CURRENT  75 

While  many  different  kinds  of  indicating  instruments  are  in 
use,  each  having  its  particular  field  of  application,  those  generally 
employed  in  direct  current  work  are  of  the  " moving  coil"  type, 
and  are  the  only  ones,  which  will  be  considered  here.  The 
working  parts  of  instruments  of  this  class  are  the  same  in  both  volt- 
meters and  ammeters,  the  differences  between  them  being  only 
in  the  method  of  connection.  TJie  instrument  proper  is,  in  reality, 
a  low  sensibility,  portable  D' Arson val  galvanometer,  consisting 
of  a  coil  of  fine  wire,  well-balanced,  and  pivoted  between  the  poles 


FIG.  41. — Working  parts  of  Weston  ammeter. 

of  a  strong,  permanent  horse-shoe  magnet.  The  magnetic  flux 
through  the  coil  is  increased  by  placing  within  it  a  cylindrical  iron 
core,  while  the  air  gap  is  further  reduced  by  pole  tips  shaped  in 
such  a  manner  as  to  make  the  field  as  nearly  radial  as  possible, 
with  respect  to  the  axis  of  the  coil.  In  this  way,  the  torque  acting 
upon  the  coil,  when  traversed  by  a  current,  is  independent  of  its 
angular  position,  the  condition  necessary  for  equal  scale  divisions. 
The  current  is  led  to  and  from  the  coil  by  spiral  springs,  which 
furnish  also  the  opposing  torque.  The  current  sensibility  of  such 
an  instrument  is  such  that  a  few  thousandths  of  an  ampere,  or 
less,  will  give  a  full  scale  deflection;  and  since  the  resistance  of  the 
instrument  is  low,  a  few  millivolts  across  its  terminals  will  furnish 


76 


ELECTRICITY  AND   MAGNETISM 


this  current.     Figure  41   shows  the  construction  of  a  Weston 
ammeter. 

When  it  is  desired  to  construct  an  ammeter,  the  instrument 
G  is  provided  with  a  shunt,  S,  as  shown  in  Fig.  42.  The  shunt, 
which  carries  the  current  to  be  measured,  has  a  resistance  (always 
low)  such  that  it  gives,  when  carrying  the  maximum  current  for 
which  it  is  designed,  a  fall  of  potential  across  its  terminals  equal 
to  that  required  for  full  scale  deflection  of  the  instrument.  For 
example,  suppose  50  millivolts  are  required  for  full  scale  deflec- 
tion, and  an  ammeter  reading  to  25  amperes  is  desired ;  the  resis- 
tance of  the  shunt  must  be 

.050 


R  = 


25 


=  .002  ohms 


By  the  law  of  shunts,  the  current  through  the  instrument  (neg- 
lected in  the  above  calculation)  is  proportional  to  that  through 
the  shunt ;  and  if  the  scale  is  divided  into  25  equal  parts,  we  have 
an  ammeter  of  the  desired  range. 


FIG.  42. — Internal  connections 
for  ammeter. 


FIG.  43. — Internal  connections 
for  voltmeter. 


The  same  instrument  may  be  used  as  a  voltmeter,  if,  instead 
of  the  shunt,  it  is  connected  in  series  with  a  large  resistance  R, 
Fig.  43,  of  such  a  value  that,  when  the  maximum  voltage  to  be 
measured  is  impressed  across  the  outside  terminals  of  G  and  R, 
the  drop  across  the  instrument  is  that  required  for  full  scale  deflec- 
tion. For  example,  suppose  the  instrument,  as  above,  requires  50 
millivolts  for  full  scale  deflection,  that  it  has  a  resistance  of  10 
ohms,  and  that  it  is  desired  to  construct  a  voltmeter  reading  to 
100  volts.  By  Ohm's  law,  R,  is  given  by  the  following  equation: 

.0501        10 
99.95       R 
Whence 


R 


10  X  99.95 
.05 


=  19,990  ohms 


MEASUREMENT  OF  CURRENT  77 

Since  the  current  through  the  instrument  is  proportional  to  the 
external  voltage  impressed,  if  the  scale  is  divided  into  100  equal 
parts,  we  have  the  voltmeter  required.  In  some  instruments, 
e.g.,  Weston,  especially  for  low  ranges,  the  shunts  and  series 
resistances,  or  multipliers,  as  they  are  generally  called,  are  placed 
within  the  case  and  cannot  be  s£e"n;  while  in  others,  e.g.,  Siemens 
and  Halske,  and  R.  W.  Paul,  they  are  mounted  outside  the  case 
and  are  detachable.  The  latter  have  the  advantage  of  being 
interchangeable,  so  that  the  same  instrument,  when  provided 
with  a  series  of  shunts  and  multipliers  of  appropriate  values, 
may  serve  either  as  a  voltmeter  or  as  an  ammeter  with  any  num- 
ber of  ranges  for  each. 

60.  Experiment  9.  Electrical  Adjustment  of  an  Ammeter  and  a 
Voltmeter. — It  is  the  purpose  of  this  exercise  to  illustrate  the 
fundamental  principles  of  construction  and  operation  of  moving 
coil  ammeters  and  voltmeters.  For  this  purpose,  a  Weston 
switch-board  type  instrument,  with  transparent  case,  has 
been  provided  with  an  ad  justable  external  shunt  and  series 
resistance.  It  is  to  be  standardized  and  tested,  first  as  an 
ammeter,  and  then  as  a  voltmeter.  In  order  to  accomplish 
this,  it  is  necessary  to  know  three  things  concerning  the  instru- 
ment: (1)  Resistance;  (2)  current  sensibility;  (3)  millivolts  for 
full  scale  deflection. 

1.  The  resistance  of  the  instrument  may  be  obtained  directly 
by  means  of  a  Wheatstone  bridge.     Set  the  ratio  coils  Fig.  15 
with  100  ohms  in  the  right-hand  bank  and  10,000  in  the  left.     Be 
careful  to  connect  the  instrument  so  that  the  pointer  moves  for- 
ward when  operating  the  bridge. 

2.  To  find  the  current  sensibility  of  the  instrument,  which  is 
defined   as  the  current  for  unit  scale  deflection,  connect  it  in 
series  with  an  adjustable  known  resistance  and  a  cell  whose 
E.M.F.  has  been  determined.     In  all  the  tests  to  be  carried  out, 
remember  that  the  instrument  is  very  sensitive,  requiring  but  an 
exceedingly  small  current  for  full  scale  deflection.     Accordingly, 
a  resistance  of  at  least  1,000  ohms  should  be  included  before  the 
circuit  is  closed.     Determine  the  resistances  corresponding  to 
five  different  indications  of  the  instrument  distributed  uniformly 
across   the   scale,  and  by  Ohm's  law,  compute  the  current  for 
unit  deflection.     The  E.M.F.  of  the  cell  may  be  obtained  by 
means  of  a  low  range  voltmeter. 

3.  The  voltage  for  full  scale  deflection  is  given  at  once  by  Ohm's 


78 


ELECTRICITY  AND  MAGNETISM 


law  as  the  product  of  the  resistance,  the  current  sensibility,  and 
the  number  of  scale  divisions. 

Part  I.  Ammeter. — It  is  required  to  construct  an  ammeter  of 
range  0-5  amperes,  from  the  instrument  and  adjustable  shunt. 
From  Ohm's  Law,  find  the  resistance,  which,  when  carrying  5 
amperes,  gives  a  potential  drop  across  its  terminals  equal  to  the 
voltage  required  for  full  scale  deflection  of  the  instrument. 
Measure  the  total  resistance  of  the  adjustable  shunt  by  means  of 
the  bridge  used  above,  correcting  for  the  leads,  and  find  what 
length  of  wire  is  necessary  for  the  required  shunt  resistance. 


Shunt 


S.A. 


-WAAAA/WW 


FIG.  44. — Connections  for  testing  improvised  ammeter. 

Now  connect  the  instrument,  as  shown  in  Fig.  44,  where  SA 
is  a  standard  ammeter  and  B,  a  storage  battery  of  6  volts,  setting 
the  shunt  at  the  computed  value.  Check  your  ammeter  against 
the  standard  ammeter  at  8  or  10  points  uniformly  distributed 
across  the  scale.  Now  compute,  as  above,  the  shunt  resistance 
required  in  order  that  your  ammeter  may  have  a  range  of  0-2.5 
amperes,  and  test  it  in  the  same  manner. 

Part  II.  Voltmeter. — It  is  required  to  construct  a  voltmeter 
of  range  0-50  volts,  from  the  instrument  and  an  adjustable  series 
resistance  used  as  a  multiplier.  From  Ohm's  law,  compute  the 
resistance  which,  when  placed  in  series  with  the  instru- 
ment, will  give  the  potential  drop  across  it  necessary  for 
full  scale  deflection,  when  50  volts  are  impressed  across  the 
instrument  and  multiplier.  Connect  the  apparatus,  as  shown 
in  Fig.  45,  placing  in  M  the  computed  resistance.  B  is  a  storage 
battery  of  50  volts,  SV  a  standard  voltmeter,  and  PD  a  potential 
dividing  rheostat  of  several  hundred  ohms,  by  means  of  which 


MEASUREMENT  OF  CURRENT 


79 


any  voltage  between  0  and  50  may  be  impressed  across  the  instru- 
ments. Check  your  voltmeter  against  the  standard  at  8  or  10 
points  evenly  distributed  across  the  scale. 

Report.— 1.  Make  a  sketch  of  the  instrument  describing  in 
detail  the  essential  working  parts. 

2.  Outline  the  general  principles  involved  in  adapting  it  to 
measure  currents  and  potential  differences. 

3.  Give  in  full  your  data  and  computations  for  shunts  and 
multiplying  resistances. 


M 


FIG.  45. — Connections  for  testing  improvised  voltmeter 


4.  Give    data    and    curves  for  your  ammeter  and  voltmeter 
calibrations. 

5.  In  calculating  the  resistance  of  the  shunt,  in  Part  I,  the 
current  through  the  instrument  was  neglected.     Compute  the 
error  thus  made. 

61.  Measurement  of  Current  by  the  Potentiometer. — Since 
the  potentiometer  measures  potentials  only,  current  measure- 
ments made  by  it  must  necessarily  be  indirect.  For  this  purpose, 
use  is  made  of  a  carefully  standardized  resistance  capable  of 
carrying  the  current  to  be  measured  without  appreciable  heating. 
The  potentiometer  measures  the  fall  of  potential  across  its  termi- 
nals produced  by  the  current,  which  is  then  determined  by  Ohm's 
law.  If  the  resistance  has  some  decimal  value,  the  value  of  the 
current  will  have  the  same  significant  figures  as  the  potential 
drop  across  it.  Accordingly,  if  the  potentiometer  has  been  made 
direct  reading  for  voltage,  it  will  indicate  currents  directly  also. 


80 


ELECTRICITY  AND  MAGNETISM 


Resistances  for  this  purpose  must  be  provided  with  two  pairs  of 
binding  posts,  one  for  current  and  the  other,  for  potential.  The 
potential  leads  are  soldered  securely  to  the  posts  between  the 
current  terminals  and  the  effective  resistance  is  only  that  between 
the  points  to  which  they  are  attached.  Errors  from  imperfect 
connections  are  thus  eliminated.  Such  resistances  should  be 
placed  in  an  oil  bath  to  keep  the  temperature  constant.  The 
largest  resistance  giving,  for  the  desired  current,  a  potential 
difference  within  the  range  of  the  potentiometer  should  be  used. 
62.  Experiment  10.  Calibration  of  an  Ammeter  by  Potentio- 
meter and  Standard  Resistance.1 — Connect  the  apparatus,  as 
shown  in  Fig.  46.  AM  is  the  ammeter  to  be  tested,  B  a  storage 
battery  of  10  volts,  S  a  rheostat  for  controlling  the  current,  and 


— L-vWWWWV 


I VVVWVWW\ 


FIG.  46. — Connections  for  standardizing  an  ammeter. 

R  a  standard  oil-cooled  resistance  provided  with  current  and 
potential  terminals.  The  leads  ab  are  to  be  connected  to  the 
potentiometer.  In  connecting  up  the  potentiometer  and  stand- 
ardizing the  current  through  it,  follow  the  directions  for  the 
particular  type  of  instrument  given  in  Chap.  IV.  After  the 
potentiometer  has  been  adjusted,  cause  such  a  current  to  flow 
in  the  ammeter  circuit  as  will  produce  about  one-tenth  full  scale 
deflection  and  measure  the  fall  of  potential  across  R  by  means  of 
the  potentiometer.  The  resistance  R  and  the  ammeter  carry  the 
same  current,  since  no  current  flows  through  a  and  b  at  the  point 
of  balance.  The  current  through  the  ammeter  is  equal  to  the 

1  JANSKY,  Electrical  Meters,  chap.  V. 
KARAPETOFF,  Experimental  Engineering,  vol.  I,  pp.  51-55. 


MEASUREMENT  OF  CURRENT  81 

reading  of  the  potentiometer  divided  by  R.  Since  R  has  a 
decimal  value,  it  is  merely  a  question  of  properly  pointing  off  this 
indication.  In  a  similar  manner,  check  the  ammeter  at  8  or  10 
points  distributed  uniformly  across  the  scale.  The  balance 
against  the  standard  cell  should  frequently  be  tested  and  any 
variations  in  the  potentiometei;  current  compensated. 

Record  ammeter  readings,  potentiometer  settings,  and  the 
value  of  R.  Note  carefully  the  zero  reading  of  the  ammeter 
before  beginning  the  test  and  again  at  the  end,  after  the  pointer 
has  been  deflected  for  some  time,  to  see  if  there  is  any  elastic 
fatigue  in  the  springs.  With  about  two-thirds  full  scale  deflec- 
tion, place  the  instrument  in  a  vertical  position  to  test  the 
accuracy  with  which  the  moving  system  is  balanced.  Bring 
another  instrument  near  this  one  to  see  if  there  is  any  effect  due 
to  external  magnetic  fields.  Tap  the  instrument  gently  with 
the  finger  to  see  if  the  bearing  friction  is  large.  Does  the  pointer 
swing  past  its  final  indication  when  a  current  is  suddenly  thrown 
on?  Record  changes  in  reading  in  all  of  the  above  cases. 

Report. — 1.  Compute  the  differences  between  the  readings  of 
the  instrument  and  true  amperes. 

2.  Plot    these    corrections    as    ordinates    against    ammeter 
readings   as   abscissas.     Draw   straight  lines   connecting  these 
points. 

3.  State    your    findings    regarding  the  imperfections  of  the 
instrument. 

4.  Would  it  indicate  on  alternating  currents? 


CHAPTER  VI 
MEASUREMENT  OF  POWER 

63.  Wattmeters.1 — Whenever    a    current   flows   in  a  circuit, 
there  is  a  certain  amount  of  energy  consumed  by  the  circuit,  and 
any  instrument  which  measures  the  rate  at  which  energy  is 
consumed  is  called  a  wattmeter,  from  the  fact  that  electrical 
power  is  generally  measured  in  watts.     Three  kinds  of  watt- 
meters are  in  common  use;  namely,  indicating,  recording,  and 
integrating.     Instruments  of  the  first  kind  show  the  power  that 
is  being  consumed  at  any  instant;  those  of  the  second  kind  make 
a  permanent  record  on  a  revolving  dial  of  the  power  consumption 
during  a  given  period  of  time;  while  those  of  the  third  kind  show 
the  total  energy,  that  is,  the  integral  of  the  power  times  the 
time,  delivered  to  a  circuit  during  a  definite  period.     Instruments 
of  the  first  kind  only  will  be  considered  here,  and  of  the  various 
types    in    use,     only    one    will    be    discussed,     namely,     the 
electrodynamometer  type. 

64.  Use  of  an  Electrodynamometer  for  the  Measurement  of 
Power. — In  case  a  steady  current  is  flowing  through  a  circuit, 
the  power  is  given  by  the  product  of  the  current  and  the  fall  of 
potential  across  the   circuit,   or 

Watts  =  Amperes  X  Volts 

The  watts  may,  therefore,  be  measured  by  simultaneously  reading 
an  ammeter  and  a  voltmeter.  If,  however,  a  single  instrument 
can  be  devised  which  will  give  indications  proportional  to  both 
current  and  voltage,  it  will  automatically  indicate  their  product, 
and  may  be  calibrated  to  read  watts  directly.  In  the  discussion 
of  the  electrodynamometer,  it  was  pointed  out  that  the  torque  is 
proportional  to  the  current  in  both  the  fixed  and  movable  coils, 
and,  therefore,  to  their  product.  Accordingly,  if  one  of  the  coils 
can  be  made  to  function  as  an  ammeter  and  the  other  as  a  volt- 
meter, the  instrument  will  be  a  wattmeter.  For  this  purpose, 
the  fixed  coil  is  made  of  a  few  turns  of  heavy  wire  and  is  connected 

1  JANSKY,  Electrical  Meters,  chap.  X. 
KARAPETOFF,  Experimental  Engineering,  vol.  I,  chap.  IV. 
Electrical  Meterman's  Handbook. 

82 


MEASUREMENT  OF  POWER 


83 


in  series  with  the  circuit  like  an  ammeter,  while  the  movable  coil 
is  made  of  a  great  many  turns  of  fine  wire  having  a  high  resistance 
and  is  connected  across  the  circuit  like  a  voltmeter  and  carries  a 
current  proportional  to  the  voltage.  The  torque  is  proportional, 
therefore,  to  amperes  times  volts,  hence,  to  watts.  This  is  the 
principle  underlying  the  Weston  Indicating  wattmeter,  the 
connections  for  which  are  shown  .in  Fig.  47.  A  and  B  are  series 
coils  consisting  of  a  few* turns  of  heavy  wire  through  which  the 
total  current  flows,  while  C  is  a  voltage  coil  of  many  turns  of  fine 
wire.  It  is  connected  across  the  load  at  the  points  H  and  K,  and 
is  mounted  so  as  to  turn  about  an  axis  through  its  geometrical 
center  perpendicular  to  the  plane  of  the  paper.  Attached  to  the 
axle  carrying  this  coil,  is  a  pair  of  spiral  springs,  not  shown  in  the 


FIG.  47. — Schematic  diagram  for 
Weston  wattmeter. 


FIG.  48. — Diagram  showing  com- 
pensating and  multiplying  coils  for 
Weston  wattmeter. 


figure,  whose  restoring  torque,  as  the  coil  is  rotated,  opposes  that 
due  to  the  electrodynamic  action  of  the  currents.  They  serve 
also  as  leads  to  and  from  the  coil.  The  scale  is  so  divided  that 
the  instrument  indicates  watts  directly. 

The  readings  of  such  an  instrument  are  subject  to  an  error  due 
to  the  power  consumed  by  the  coils  themselves.  An  inspection 
of  Fig.  47  shows  that  the  current  passing  through  the  coils  A 
and  B  is  the  sum  of  the  load  current  and  that  carried  by  the  coil 
C,  hence  the  reading  must  be  too  large  by  the  PR  loss  in  this  coil. 
If  it  is  attempted  to  overcome  this  by  connecting  the  voltage 


84  ELECTRICITY  AND  MAGNETISM 

terminals  on  the  "line"  side  of  the  current  coils,  the  registered 
voltage  will  be  too  large  by  the  drop  across  the  current  coils  thus 
again  making  the  reading  too  large.  To  overcome  this  difficulty, 
a  compensating  coil  M  is  provided  as  shown  in  Fig.  48,  which  is 
usually  placed  inside  A  and  B  and  so  connected  that  its  magnetic 
effect  weakens  their  fields,  thus  automatically  correcting  the 
reading  of  the  instrument.  If  the  wattmeter  is  to  be  calibrated 
by  using  separate  sources  of  current  and  potential,  this  compensa- 
tion is  not  necessary,  and  a  separate  binding  post  F  is  provided, 
marked  Ind.  (Independent)  on  the  instrument.  This  circuit 
includes  a  resistance  r  equal  to  that  of  the  compensating  coil, 
thus  making  the  resistance  between  C  and  F  equal  to  that 
between  C  and  L.  The  series  resistance  R  is  used  as  a  multiply- 
ing resistance  in  exactly  the  same  manner  as  the  multiplier  in  an 
ordinary  D.C.  voltmeter.  For  example,  if  R  is  equal  to  the 
resistance  of  the  movable  coil,  the  potential  difference  across  it 
will  be  equal  to  that  across  the  coil,  and  if  the  instrument  is 
calibrated  without  R  in  circuit,  when  R  is  included,  the  readings 
should  be  multiplied  by  the  factor  two. 

65.  Experiment  11.  Calibration  of  a  Wattmeter. — Wattmeters 
are  calibrated  on  direct  currents  and  may  be  used  on  alternating 
currents  as  well  as  direct.  Separate  sources  of  current  and 
electromotive  force  are  generally  used  for  purposes  of  calibration 
since  instruments  of  large  capacity  may  then  be  standardized 
with  a  comparatively  small  expenditure  of  power.  Connect 
the  apparatus  as  shown  in  Fig.  49,  where  WM  is  the  wattmeter 
which  is  to  be  calibrated.  B  is  ten-volt  storage  battery  furnish- 
ing the  current  which  is  controlled  by  the  rheostat  R  and  read 
by  the  ammeter  AM.  C  is  another  storage  battery  furnishing 
the  potential  which  is  controlled  by  the  voltage  regulating  rheo- 
stat PS  and  read  by  the  voltmeter  VM.  Since  the  field  due  to 
the  coils  of  the  instrument  is  small,  extraneous  fields,  such  as 
those  of  the  earth  or  large  currents,  near-by  instruments  with 
permanent  magnets,  etc.,  may  cause  errors  as  large  as  several 
per  cent.  Hence  it  is  necessary,  when  using  this  type  of  watt- 
meter on  direct  currents,  to  reverse  both  potential  and  current 
leads  and  average  the  two  readings.  Make  two  calibrations. 
First,  hold  the  current  constant  and  vary  the  voltage  so  as  to 
check  the  instrument  at  eight  or  ten  points  uniformly  distributed 
across  the  scale.  Next  hold  the  voltage  constant  and  vary  the 
current,  checking  approximately  the  same  points  as  before. 


MEASUREMENT  OF  POWER 


85 


Record  volts,  amperes,  and  indicated  watts,  both  direct  and 
reversed,  in  all  cases.  With  about  two-thirds  full  scale  deflec- 
tion, bring  an  instrument  with  a  permanent  magnet  near  the 
wattmeter  and  note  the -effect  on  the  reading.  Place  the  watt- 
meter pointing  in  various  directions  and  note  any  changes  due 
to  the  earth's  magnetic  field.  Stand  the  instrument  in  a  vertical 
position  and  note  any  error  due  to  imperfect  balancing  of  the 
moving  system.  Change  the  voltage  terminal  from  the  post  F, 
marked  "Ind."  to  L  and  note  the  difference,  which  is  the  correc- 
tion for  internal  energy  consumption. 


FIG.  49. — Connections  for  calibrating  a  wattmeter. 

Report. — 1.  Compute  true  watts  from  the  average  products 
of  amperes  and  volts. 

2.  Plot  corrections  as  ordinates  against  wattmeter  readings  as 
abscissas.     Do  the  two  curves  (a)  with  current  constant,  and  (6) 
with  voltage  constant,  coincide? 

3.  State  your  findings  regarding  internal  energy  consumption, 
effects  of  extraneous  magnetic  fields,  balancing  of  system,  etc. 

4.  Why  are  the  scale  divisions  in  this  wattmeter  unequal  and 
those  of  the  D.C.  voltmeter  and  ammeter  equal? 


CHAPTER  VII 
MEASUREMENT  OF  CAPACITANCE 

66.  Condensers.— When  a  body  is  charged  with  a  quantity  of 
electricity  Q,  the  potential  V  which  the  body  acquires  is  propor- 
tional   to    Q.     With   a    given  charge,    however,    the    potential 
depends  also  upon  certain  conditions  of  the  body  such  as  size, 
shape,  surrounding  medium,  presence  of  other  charges,  etc.     The 
relation  between  charge  and  potential  is  given  by  the  equation 

Q  =  CV  (1) 

where  C  is  a  constant  depending  upon  the  conditions  of  the  body, 
and  is  called  the  "  Capacitance  "  of  the  body.  It  is  the  ratio  of 
the  charge  to  the  potential  and  is  numerically  equal  to  the  charge 
when  the  potential  is  unity.  The  practical  unit  of  capacitance  is 
the  farad.  A  body  is  said  to  have  a  capacitance  of  one  farad 
when  a  charge  of  one  coulomb  raises  its  potential  by  one  volt. 
The  farad  is  too  large  a  unit  for  practical  purposes,  however,  and 
it  is  customary  to  take  the  millionth  part  of  this,  called  the  micro- 
farad, as  a  working  unit.  Any  device  by  which  it  is  possible  to 
cause  a  large  quantity  of  electricity  to  exist  under  a  relatively 
small  potential  is  called  a  condenser.  Such  devices  usually 
consist  of  thin  conducting  plates,  placed  close  together,  but 
insulated  electrically  by  thin  sheets  of  some  good  dielectric 
material.  If  a  positive  charge  is  placed  upon  one  plate  and  a 
negative  upon  the  other,  the  neutralizing  effect  of  each  on  the 
other,  due  to  their  close  proximity,  causes  the  potential  difference 
between  them  to  be  very  much  reduced  over  what  it  would  have 
been  if  they  were  far  apart. 

67.  Grouping   of   Condensers. — Condensers,  like  resistances, 
may  be  joined  either  in  series  or  in  parallel  and  used  as  a  single 
condenser.     Figure   50   represents   three    condensers   joined   in 
parallel.     Let  Ci,  Cz,  Cz  represent  their  individual  capacitances, 
#i>  #2,  #3  their  charges;  and  E,  the  difference  of  potential  across 
their  terminals.     Calling  Q  the  total  quantity  of  electricity  stored 
in  the  group,  which  would  be  obtained  if  they  were  discharged, 
we  have 

Q  =  qi  +  02  +  93  (2) 

86 


MEASUREMENT  OF  CAPACITANCE 


87 


If  C  is  the  resultant  capacitance  of  the  group,  we  have,  from 
definition, 

Q  =  CV  =  CiV  +  C2V  +  C,V  (3) 

or 

C  =  Ci  +  C2  +  C3  (4) 

For  condensers  connected  in  parallel,  the  resultant  capacitance 
is  the  sum  of  the  individual  capacitances  of  the  group.  The 
capacitance  of  n  similar  condensers  thus  joined  is  n  times  the 
capacitance  of  a  single  condenser.  Figure  51  represents  three 
condensers  connected  in  series.  As  before,  let  Ci,  C2,  Ca  represent 


FIG.  50. — Condensers 
joined  in  parallel. 


FIG.  51. — Condensers  joined  in  series. 


the  values  of  the  individual  condensers;  qit  q2,  qs  their  charges, 
and  vi,  v2,  v-i  the  potential  differences  across  each.  It  is  evident 
from  the  figure  that 

E  =  vi  +  v2  +  vs  (5) 

Calling  C  the  resultant  capacitance  of  the  group,  and  Q  the  total 
charge,  we  have,  from  definition, 

F  -  Q  -    21  j    ?2  4    V*  fp\ 

&  ~"7>~/T~r?^     TTT  \v) 

U  i  L/  2  v-'S 

A  simple  relation  exists  between  these  charges.  We  have  tacitly 
assumed  that  the  condensers  were  uncharged  before  connection. 
Suppose  a  unit  charge  passes  from  the  battery  to  the  outer  coating 
of  Ci.  A  negative  charge  will  then  be  induced  on  the  inner  coat- 
ing and  a  positive  unit  charge  repelled  from  it  which  will  charge  the 
outer  coating  of  C2  and  induce  a  negative  unit  charge  on  its  inner 
coating  and  so  on.  The  next  unit  charge  from  the  battery  will 
do  the  same.  It  is  evident  then  that  the  charge  for  each  conden- 
ser is  the  same,  no  matter  what  its  capacitance  and  that  the  total 
charge  which  may  be  obtained  from  the  group  on  discharge  is 
the  same  as  the  charge  from  any  one  condenser.  That  is,  the 


88 


ELECTRICITY  AND  MAGNETISM 


positive  charge  on  the  outer  coating  of  C\  neutralizes  the  charge 
on  the  inner  coating  of  C3  and  similarly  for  the  other  condensers 
of  the  group.  Thus  we  have 

Q  =  qi  =  qz  =  ?3  (7) 

and 


1=1+1+1 

C  C  i  C  2  (^3 


(8) 


For  condensers  joined  in  series,  the  reciprocal  of  the  resultant 
capacitance  is  the  sum  of  the  reciprocals  of  the  individual  capaci- 
tances. The  resultant  capacitance  of  n  similar  condensers  so 

joined  is  -  times  the  capacitance  of  a  single  condenser. 

ft 

68.  Standard  Condensers. — Standard  condensers  are  made  of 
sheets  of  tin  foil  separated  by  mica,  alternate  sheets  of  foil 
being  joined  as  shown  in  Fig.  50,  and  the  whole  finally  imbed- 
ded in  solid  paraffin.  Figure  52  shows  the  connections  for 


p 

EARTH 

j 

! 

|| 

i  i 

| 

j 

A 

M 
|| 

3 

C 

D 

if!] 

u 

E 

05                 .05                   .2                   ,Z                  .5 
CONDENSER 

FIG.  52. — Connections  for  subdivided  standard  condenser. 

one  of  the  subdivided  standard  condensers  used  in  the  labora- 
tory. One  side  of  each  of  the  sections  shown  by  the  dotted 
lines  is  joined  to  a  heavy  bar  marked  "Earth"  and  the  other 
sides  to  one  of  the  blocks.  Another  bar  marked  "Condenser'' 
is  mounted  opposite,  and  each  bar  is  connected  to  a  binding  post. 
When  it  is  desired  to  use  a  certain  capacity,  e.g.,  2  MF,  place  a 
plug  in  the  socket  between  C  and  the  lower  bar.  If  .7  MF  is 
desired,  place  another  plug  between  E  and  this  bar.  Similarly 
for  the  various  other  possible  connections.  When  a  section  is 
not  in  use,  it  should  be  short  circuited  by  placing  a  plug  in  the 
socket  between  the  upper  bar  and  the  corresponding  block.  Care 
should  be  taken  never  to  place  plugs  at  both  ends  of  any  block  as 


MEASUREMENT  OF  CAPACITANCE 


89 


that  would  short  circuit  the  entire  condenser,  possibly  injuring 
apparatus  to  which  it  is  connected. 

Another  method  of  assembling  subdivided  standard  condensers 
is  to  join  the  units,  not  between  the  central  lugs  and  one  of  the  bus 
bars  as  shown  in  Fig.  52,  but  to  connect  them  between  the  lugs 
as  shown  in  Fig.  53.  Thus  between  A  and  B  there  is  .05  micro- 


co 


-II- 


-II- 


MICRO-FARAD 


FIG.  53. — Alternate  method  of  connecting  condenser  units. 

farads,  between  B  and  C,  .05,  etc.  One  more  lug  is  required  in 
this  case  To  connect  all  the  units  in  parallel,  plugs  are  inserted 
in  alternate  holes  on  each  side,  but  staggered.  The  method  has 
the  advantage  of  permitting  series  grouping  of  the  units,  thus 
giving  a  greater  number  of  values  of  capacitance  for  a  given 
number  of  units.  A  subdivided  standard  condenser  is  shown  in 
Fig.  54. 


FIG.  54. — Subdivided  standard  condenser. 

69.  Comparison  of  Condensers. — The  capacitance  of  an 
unknown  condenser  may  be  found  by  comparing  it  with  a  stan- 
dard condenser.  A  ready  means  of  doing  this  which  gives  results 
sufficiently  accurate  for  many  purposes  is  to  charge  both  the 
unknown  and  the  standard  to  the  same  potential  difference  and 
discharge  each  in  succession  through  a  ballistic  galvanometer. 
The  set-up  for  this  purpose  is  shown  in  Fig.  9.  Let  Ci  be  the 
unknown  and  C2  the  standard  condenser.  First  insert  Ci,  then 
charge  and  discharge  several  times,  taking  the  average  deflection 


90 


ELECTRICITY  AND  MAGNETISM 


which  we  will  call  d\.  From  the  definition  of  capacitance  we 
have,  as  the  charge  in  the  condenser. 

Qi  =  Ci7,  (9) 

and,  since  the  deflection  of  a  ballistic  galvanometer  is  propor- 
tional to  the  charge  passed  through  it, 

Q1  =  C1V  =  Kdl  (10) 

Now  replace  the  unknown  by  the  standard  condenser  and  charge 
and  discharge  as  before.  In  a  similar  manner,  we  have 

Q2  =  C2F  =  Kd*  (11) 

Dividing  equation  (10)  by  (11), 

—  -  -  fl^ 

C2       d* 

70.  Bridge  Method  for  Comparing  Two  Condensers.1 — A  more 
accurate  comparison  of  two  condensers  may  be  made  by  means 


U 


U 


D 


f  T 

i 

H 

4    £ 

:        4 

FIG.  55. — Bridge  method  for  comparing  two  condensers. 

of  an  arrangement  similar  to  the  Wheatstone  bridge  in  which  the 
resistances  of  two  of  the  arms  are  replaced  by  the  two  condensers 
to  be  compared,  and  the  current  galvanometer  is  replaced  by  a 
1  CARHART  and  PATTERSON,  Electrical  Measurements,  pp.  213-220. 
SMITH,  Electrical  Measurements,  art.  IX. 


MEASUREMENT  OF  CAPACITANCE  91 

ballistic  galvanometer.  The  connections  are  shown  in  Fig.  55. 
Ci  and  C2  are  two  condensers  and  RI  and  Rz  two  variable  resis- 
tance boxes.  K  is  a  charge  and  discharge  key  so  connected  that, 
when  the  blade  is  pressed  down,  the  battery  B  is  connected  to 
the  bridge,  thus  charging  the  condensers  through  RI  and  R2 
to  the  potential  difference  furnished  by  the  battery.  When 
contact  is  made  on  the  upper  p&int,  the  battery  is  disconnected 
and  the  condensers  are^discharged  through  the  resistances.  No 
matter  what  the  values  of  RI  and  R%  may  be,  the  points  P  and  Q 
will  come  to  the  same  final  potentials  on  charge  and  again  on 
complete  discharge,  since,  when  no  current  is  flowing  through 
the  resistances  there  can  be  no  fall  of  potential  across  them. 
However,  during  the  charging  and  discharging  processes  there 
are  currents  through  the  resistances  and,  in  general,  there  will  be 
a  momentary  difference  of  potential  between  P  and  Q  causing  a 
deflection  of  the  galvanometer  in  one  direction  on  charge,  and 
in  the  opposite,  on  discharge.  By  properly  adjusting  RI  and 
Rz,  it  is  possible  to  make  the  potentials  at  P  and  Q  rise  and  fall 
at  the  same  rate  which  is  the  balance  condition  for  the  bridge, 
from  which  the  relation  between  the  capacitances  and  resistances 
may  be  deduced. 

Let  qi  and  qz  =  instantaneous  charges  in  Ci  and  Cj 
Let  ii  and  iz  =  instantaneous  currents  in  R  i  and  Rz 

As  in  the  ordinary  Wheatstone  bridge,  we  have 

P.D.  between  A  and  P  =  P.D.  between  A  and  Q 
P.D.  between  P  and  D  =  P.D.  between  Q  and  D 

whence 

#!  #2  /10\ 

c~ :=  c"  '  ' 

and 

•p     •  7")     •  / 1  A  \ 

Kil\   =  ri^'lz  \J-^v 

Differentiating  (13)  and  remembering  that  i  =  -4f  we  have 


Cidt 
or 


Idqi  = 

[  /~y    jt 

LzCtt 

£  (15) 

^2 


92  ELECTRICITY  AND  MAGNETISM 

Dividing  (14)  by  (15) 

RiCi  =  #2C2  (16) 

or 

Cl 


It  is  to  be  noted  that  the  ratio  of  the  capacitances  is  the  inverse 
ratio  of  the  resistances,  whereas,  in  the  bridge  method  for  resis- 
tances, it  is  the  direct  ratio. 

71.  Experiment  12.     Comparison  of  Two  Capacitances  by  the 
Bridge  Method.  —  Connect  the  apparatus  as  shown  in  Fig.  55, 
using  for  B  a  battery  of  40  volts.     SK  is  a  short  circuiting  key 
to  bring  the  galvanometer  to  rest  after  taking  an  observation. 
Use  for  Ci  a  subdivided  standard  condenser  and  for  C2  a  fixed 
condenser  of  about  one-half  micro-farad  capacity.     The  problem 
is  to  check  the  parts  of  Ci  in  terms  of  the  whole.     Set  the  plugs 
of  Ci  so  as  to  give  the  maximum  available  capacitance,  and  with 
this  as  a  standard,  obtain  several  balances  on  C2,  using  different 
values  for  RI  and  Rz  in  each  case.     Now,  taking  this  measured 
value  of  C2  as  a  standard,  determine  the  capacitance  of  each 
part  of  Ci,  making  several  independent  balances  for  each.     In 
all  cases  use  as  large  values  for  RI  and  R2  as  possible. 

Report.  —  Tabulate  your  data  in  compact  form.  Your  values 
for  the  separate  parts  of  Ci  should  add  up  to  the  total  value  indi- 
cated on  the  top  of  the  box. 

72.  Measurement  of  Small  Capacitance  by  Commutator.1  — 
The  bridge  method  just  described  is  not  suited  to  the  measure- 
ment of  small  capacitances  since  the  charging  currents  are  so 
minute    that  the  potential  drops  through  the  resistances  are 
inappreciable.     For  the  determination  of  a  small  capacitance, 
such  as  that  of  an  air  condenser  used  in  radio  work  or  that  between 
the  wires  of  a  transmission  line,  a  direct  deflection  method  may  be 
used  in  which  the  condenser  is  rapidly  charged  to  a  known  voltage 
and  then  discharged  through  a  standardized  galvanometer  by 
means  of  a  motor-driven  commutator.     If  the  interval  between 
discharges  is  small  compared  to  the  period  of  the  galvanometer, 
a  steady  deflection  results  which  is  proportional  to  the  average 
value  of  the  current. 

1  FLEMING  and  CLINTON,  Proc.  Phys.  Soc.  of  London,  vol.  18,  1901-03, 
p.  38f>. 


MEASUREMENT  OF  CAPACITANCE 


93 


Figure  56  shows  the  principle  of  the  Fleming  and  Clinton  com- 
mutator, designed  for  this  purpose,  together  with  the  wiring 
diagram.  Si  and  Sz  are  slip  rings  which  revolve  in  a  plane  per- 
pendicular to  the  paper  while  /S3  is  a  series  of  posts  alternately 
connected  to  Si  and  $2.  When  brush  3  rests  upon  a  segment 
connected  to  Si,  the  condenser  C  is  charged  to  the  potential 
difference  of  the  battery  B,  and  when  3  touches  the  succeeding 
post,  the  condenser  is  discharged 
through  the  galvanometer.  Let  n  be 
the  number  of  discharges  per  second 
and  V  the  E.M.F.  of  the  battery. 
Then  the  current  through  the  gal- 
vanometer is 

I  =  nCV  X  10-6  =  Fd       (18) 

Where  C  is  the  capacity  of  the  con- 
denser in  microfarads,  and  F,  the 
figure  of  merit  of  the  galvanometer, 
i.e.,  the  current  for  unit  deflection. 

In  designing  a  commutator  of  this 
type,  special  precautions  must  be 
taken  to  secure  good  insulation  be- 
tween posts  and  sectors.  They  are 
generally  made  with  an  air  gap  be- 
tween posts  since  metal  abraded  by 

friction  otherwise  embeds  itself  in  a  solid  dielectric  thus  giving 
a  direct  leakage  path  from  the  battery  through  the  galvano- 
meter. A  speed  counter,  mounted  on  the  shaft,  indicates  the 
number  of  revolutions. 

73.  Experiment  13.  Capacitance  by  the  Fleming  and  Clinton 
Method. — Connect  the  apparatus  as  shown  in  Fig.  56  using  for  C 
an  air  condenser  of  small  capacitance.  Drive  the  commutator 
at  speeds  sufficient  to  give  50  to  100  discharges  per  second  through 
the  galvanometer,  and  use  for  B  a  battery  with  voltage  large 
enough  to  produce  a  deflection  of  about  10  cms.  Take  special 
precautions  to  insure  good  insulation  between  the  galvanometer 
and  battery  circuits.  Use  a  number  of  different  speeds  and 
voltages  and  determine  the  value  of  C  by  eq.  (18).  Determine 
the  figure  of  merit  of  the  galvanometer  by  the  method  given 
in  Art.  23. 


FIG.  56. — Fleming  and  Clinton 
commutator. 


94  ELECTRICITY  AND  MAGNETISM 

Report. — 1.  Tabulate  observations  and  results  for  the  series  of 
measurements  taken. 

2.  Check  your  results  by  measuring  the  dimensions  of  the 
condenser,  and  computing  its  capacity  from  the  formula  for  the 
parallel  plate  condenser 

C  =  .0885  X  10~6  ^p  microfarads  (19) 

where  A  is  the  area  of  one  of  the  plates  in  square  centimeters,  d 
the  thickness  of  the  dielectric,  and  k  the  dielectric  constant 
(K  =  1  for  air). 

3.  How  do  you  account  for  the  difference  between  the  mea- 
sured and  computed  values? 


CHAPTER  VIII 

> 

MAGNETISM 

74.  General  Principles. — Magnetism  is  a  universal  property 
of  matter,  since  there  is  no  substance  which  does  not  experience 
a  ponder-motive  action  when  placed  near  the  poles  of  a  strong 
magnet,  though  in  many  cases  the  effect  is  so  weak  that  delicate 
means   are   necessary   for   its    detection.     Substances   may   be 
divided  into  two  groups,   in  accordance  with  the  manner  in 
which  they  behave  when  acted  upon  by  a  magnetic  pole;  those 
which  are  attracted  by  the  magnet  are  called  paramagnetic,  and 
those  repelled,   diamagnetic.     It  is  customary  to  add  a  third 
group,  including  iron,  nickel,  and  cobalt,  which  are  characterized 
by  the  fact  that  the  ponder-motive  action  upon  them  is  not 
proportional  to  the  strength  of  the  attracting  pole  as  in  the  case 
of  ordinary  paramagnetic  substances,  but  is  much  stronger. 

75.  Strength  of  Pole. — In  the  early  study  of  magnets,  it  was 
noticed  that  the  magnetic  property  of  a  body  is  confined  largely 
to  the  areas  about  its  ends  and  corners,  and  that  opposite  ends 
behave  differently  toward  other  magnets.     The  term  magnetic 
pole  was  given  to  the  regions  where  the  property  was  most 
pronounced  and  has  been  retained  although  it  has  been  known 
for  a  long  time  that  magnetism  is  a  volume  and  not  a  surface 
phenomenon.     A    unit   magnetic   pole   is   defined   as   one   which 
repels  an  exactly  similar  pole  at  a  distance  of  one  centimeter  in  air 
with  a  force  of  one  dyne. 

76.  Strength  of  Field. — The  space  surrounding  a  magnetic 
pole  in  which  action  upon  another  pole  can  be  detected  is  called 
a  magnetic  field,  and  is  measured,  at  any  particular  point,  by  the 
force  in  dynes  with  which  a  free  unit  positive  pole  is  acted  upon 
when  placed  at  that  point.     A  field  of  unit  strength  or  intensity  is 
one  which  will  exert  a  force  of  one  dyne  upon  a  unit  pole.     Since 
field  strength  is  thus  measured  by  force  per  unit  pole,  it  is  a 
vector  quantity;  i.e.,  it  possesses  both  magnitude  and  direction. 
Both  of  these  characteristics  may  be  represented  by  imagining 
lines  drawn  in  space  according  to  a  definite  convention;  namely, 

95 


96  ELECTRICITY  AND  MAGNETISM 

the  magnitude  of  the  field  by  drawing  as  many  lines  per  square 
"centimeter  as  the  field  has  units  of  intensity,  and  the  direction, 
by  making  these  lines  coincide  at  every  point  with  the  direction 
in  which  the  unit  measuring  pole  is  urged.  According  to  this 
convention,  if  a  sphere  of  unit  radius  is  drawn  with  a  pole  of 
strength  m  as  a  center,  there  must  pass  through  each  square 
centimeter  of  its  surface  m  lines,  giving  a  total  of  4irm  lines  from 
the  pole.  From  a  unit  pole  there  will  emanate  according  to  this 
convention,  4ir  lines  of  force. 

77.  Intensity   of   Magnetization. — Let   us   imagine   an  ideal 
permanent  bar  magnet,  of  length  L,  and  uniform  cross  section  A, 
magnetized  uniformly  and  showing,  therefore,  pole-strength  over 
the  ends  only.     That  is  to  say,  the  magnetic  lines  all  leave  one 
end,  pass  in  regular  curves  through  the  outside  space,  and  enter 
the  other  end  with  no  lines  entering  or  leaving  on  the  side,  as  in 
any  real  magnet.     Let  the  strength  of  pole  be  m.     The  pole 

strength  per  unit  area,  -r,  is  defined  as  the  intensity  of  magnetiza- 
tion and  is  generally  represented  by  the  letter  /. 

78.  Magnetic  Moment. — Imagine  the  ideal  magnet  mentioned 
above  placed  at  right  angles  to  a  uniform  magnetic  field  of 
strength  H.     Equal  and  opposite  forces  of  magnitude  Hm  will 
act  upon  this  magnet  producing  a  couple  of  strength  HmL. 
If  H  is  unity,  the  magnitude  of  the  couple  is  raL,  and  this  quan- 
tity,   which    is    exceedingly    important    in    treating    problems 
involving  magnets,  is  called  the  magnetic  moment,  and  is  designated 
by  the  letter  M.     The  moment  of  any  magnet  is,  then,  the  torque 
acting  upon  it  when  placed  at  right  angles  to  a  uniform  field  of 
unit  strength.     Another  definition  of  intensity  of  magnetization 
in   terms   of  magnetic   moment  may   be   obtained   as   follows: 
Since  the  volume  of  the  bar  magnet  is  LA,  we  have 

r  _  m  _  mL  _  Af  (  . 

=  A==AL  =    ~V 

Intensity  of  magnetization  is  thus  defined  as  a  volume  rather 
than  a  surface  effect. 

79.  Magnetic  Induction. — Let  us  imagine  that,  in  an  infinitely 
long,  uniform  magnetic  field  of  strength  H ,  an  iron  bar  is  placed 
with  its  axis  parallel  to  the  field.     The  bar  becomes  magnetized 
to  an  intensity  /  and  is  equivalent  to  the  ideal  magnet  considered 
above.     The    number    of   magnetic    lines    through    the    space 


MAGNETISM  97 

occupied  by  the  bar  has  been  increased  by  the  lines  of  magnetization 
due  to  the  bar.  The  total  number  of  magnetic  lines  through  the 
bar,  which  is  made  up  of  the  original  lines  and  the  lines  of  mag- 
netization, is  called  the  magnetic  flux,  and  is  generally  designated 
by  the  Greek  letter  <£.  The  number  of  lines  per  square  centi- 
meter through  the  bar  is  called-  the  magnetic  induction,  and  is 
represented  by  the  letter  B.  Thus 

"*"  Total  Flux        </> 

B  =  Induction  =  -  -  =  -  (2) 

Area  A 

The  induction  B  is  denned  in  the  following  manner:  Imagine  a 
narrow  crevasse  cut  through  the  middle  of  the  bar  at  right  angles 
to  its  axis,  and  a  unit  positive  pole  placed  within.  The  force  in 
dynes  upon  this  pole  measures  B.  The  original  field  produces  a 
force  of  H  dynes  upon  the  pole,  and  since  the  iron  is  magnetized 
to  an  intensity  /,  meaning  /  units  of  pole  strength  per  unit  area  of 
the  crevasse,  from  each  of  which  4?r  lines  emanate,  we 
have,  as  the  total  lines  per  square  centimeter  through  the  gap 
or  the  force  in  dynes  acting  upon  the  unit  pole 

B  =  H  +  47r/  (3) 

Lines  of  induction  are  continuous  throughout  the  magnetic 
circuit;  that  is,  they  never  begin  or  end  but  form  closed  paths, 
the  parts  in  the  air  being  called  lines  of  force.  If,  instead  of  the 
transverse  crevasse  we  had  bored  a  small  hole  through  the  bar 
parallel  to  the  lines  of  force  and  placed  a  unit  magnetic  pole  within, 
the  force  upon  it  would  be  the  original  strength  of  field  H  which 
has  produced  the  magnetic  induction. 

80.  Permeability  and  Susceptibility.  —  For  many  purposes  it 
is  convenient  to  define  the  magnetic  quality  of  a  given  material 
in  terms  of  the  relative  increase  in  the  number  of  lines  or  the 
intensity  of  magnetization  produced.  For  this  purpose  the  terms 
permeability  and  susceptibility  are  used.  By  permeability  is 
meant  the  ratio  of  the  induction  B  to  the  field  strength  H,  and  is 
represented  by  the  Greek  letter  /*.  That  is, 


where  B  is  the  induction  produced  in  a  given  material  when 
acted  upon  by  a  field  of  strength  H.  When  it  is  desired  to  express 
the  ability  of  a  material  to  acquire  magnetism  and  to  state  its 
condition  in  terms,  not  of  the  total  induction,  but  of  its  own 
magnetic  lines  alone,  we  use  the  term  susceptibility.  This  is 


98  ELECTRICITY  AND  MAGNETISM 

defined  as  the  ratio  of  the  intensity  of  magnetization  of  the 
specimen  to  the  magnetizing  field  in  which  it  is  placed,  and  is 
represented  by  the  Greek  letter  K.  That  is, 


A  simple  relation  exists  between  these  two  quantities.     Taking 
the  defining  equation  for  induction 

B  =  H  +  47TI  (6) 

and  dividing  through  by  H,  we  have 


or 

M  =  1  +  4™  (8) 

81.  Effects  of  the  Ends  of  a  Bar.  —  When  a  bar  is  magnetized 
longitudinally  by  placing  it  in  a  magnetic  field,  the  ends  become 
poles  which  act  upon  any  other  pole  in  the  neighborhood,  attract- 
ing or  repelling  it  according  to  the  relative  signs  of  the  poles. 
If  the  bar  lies  in  an  east-west  position,  magnetized  with  a  north 
pole  at  the  west  end,  a  unit  north  pole  lying  near  the  middle  of  the 
bar,  but  outside  it,  would  be  urged  from  west  to  east  or  in  a 
direction  opposite  to  that  in  which  the  bar  is  magnetized.  If  now 
the  unit  pole  is  placed  within  the  bar,  the  force  is  in  the  same 
direction.  Thus  the  effect  of  the  poles  is  to  produce  a  field  within 
the  bar  called  a  "  demagnetizing  field"  which  is  opposite  to  the 
direction  of  the  field  magnetizing  it.  This  effect  is  greater  the 
shorter  the  bar  is  in  comparison  to  its  diameter.  The  actual 
field  producing  magnetization  is,  accordingly,  less  than  the  field 
before  the  bar  was  introduced.  This  phenomenon  is  allowed  for 
by  computing  the  effective  field  H  from  the  equation 

H  =  Hf  -  NI  (9) 

where  H'  is  the  original  field  and  N  a  constant  depending  upon 
the  ratio  of  the  length  to  the  diameter  of  the  bar,  and  is  called 
the  "  Demagnetizing  Factor."  Tables1  for  N  may  be  found  in  the 
more  advanced  treatises  on  the  subject.  The  same  considera- 
tions hold  for  solenoids,  and  hence  it  is  necessary,  when  one 
wishes  a  solenoid  whose  field  may  be  computed  readily  from  its 
dimensions,  to  make  it  long  in  comparison  to  its  diameter.  If 
one  uses  a  ring  solenoid,  or  a  test  specimen  in  the  form  of  a  ring, 
1  Du  Bois,  The  Magnetic  Circuit,  p.  41. 


MAGNETISM  99 

this  correction  is  unnecessary  since  there  are  no  free  poles  to 
produce  disturbing  effects  of  this  character. 

82.  The  Magnetic  Circuit.  —  In  treating  such  phenomena  as  the 
conduction  of  heat  and  the  flow  of  electricity,  one  makes  use  of  a 
general  law  in  which  the  magnitude  of  the  effect  is  given  as  the 
ratio  of  a  driving  force  divided  by  an  opposition  factor  dependent 
upon  the  properties  of  the  medium  in  which  the  action  takes 
place.  For  example,  the  heat  current  Q,  i.e.,  the  quantity  of  heat 
passing  per  unit  time  any  cross  section  of  a  conductor  of  length  L 
and  cross  sectional  area  A  ,  when  the  temperature  at  the  ends  are 
ti  and  h,  is  given  by  the  expression 

Q  =  (10) 


where  r  is  a  constant  defining  the  ability  of  the  medium  to  conduct 
heat,  r  is  called  the  specific  thermal  conductivity  and  is  numeri- 
cally equal  to  the  quantity  of  heat  passing  through  a  centimeter 
cube  of  the  material,  per  unit  time,  when  a  difference  of  tempera- 
ture of  one  degree  is  maintained  across  its  faces.  Similarly,  the 
electrical  current  flowing  in  the  above  conductor  when  its  ends  are 
maintained  at  electrical  potentials  V\  and  V%,  is  given  by 


"" 


A  CA 

where  C  =  -  is  called  the  specific  electrical  conductivity  of  the 

material  and  is  numerically  equal  to  the  current  flowing  through  a 
centimeter  cube  when  unit  difference  of  potential  is  maintained 
across  its  faces.  Its  reciprocal  p  is  called  the  specific  resistance, 
and  is  the  resistance  of  the  centimeter  cube.  This  equation  is 
called  Ohm's  law  and  is  written 

Electromotive  Force 
Current  =  -    —  ~  —  r— 

Resistance 

In  an  analogous  manner  it  is  convenient,  for  purposes  of  calcula- 
tion, to  regard  the  region  in  which  a  magnetized  state  exists  as 
being  the  seat  of  a  magnetic  flow.  The  magnetic  lines  of  induc- 
tion are  the  stream  lines  along  which  the  flow  takes  place,  and 
since  magnetic  lines  are  closed  paths,  the  lines  of  flow  are  closed 
circuits.  Materials  are  then  classified  as  good  or  bad  magnetic 
conductors  according  to  the  ease  with  which  they  are  magnetized. 


100  ELECTRICITY  AND  MAGNETISM 

To  make  the  analogy  clear,  consider  a  specimen  of  magnetic 
material  in  the  form  of  an  anchor  ring,  wound  uniformly  with  wire 
through  which  a  current  is  flowing,  as  shown  in  Fig.  57.  We  wish 
to  compute  the  total  magnetic  flux  produced  in  the  ring  when  a 
given  current  is  flowing. 

Let  N  =  total  number  of  turns 

L  =  mean  length  of  magnetic  lines 
A  =  area  of  cross  section  of  ring 
B  =  magnetic  induction  in  ring 
M  =  permeability 
I  =  strength  of  current 

As  a  direct  consequence  of  the  defini- 
tion of  the  electromagnetic  unit  of 
current,  it  is  shown  in  elementary 
textbooks  that  the  work  done  in  carry- 

The  magnetic  circuit.    ing  ft  ^  magnetic  pole  Qnce  around 

a  current  of  strength  I  E.M.U.'s,  is  4?rl  ergs.  The  field  strength 
within  the  ring  solenoid  may  be  obtained  from  the  fact  that  the 
work  done  in  taking  a  unit  pole  around  this  magnetic  circuit  is 

Work  =  HL  =  4irNI  (12) 

since  the  pole  is,  in  reality,  carried  N  times  around  the  current. 
Whence 


a  =  (is) 

Li 

If  I  is  expressed  in  amperes  instead  of  electromagnetic  units, 

H  =  —  (14) 

10L 

From  the  above  definitions,  the  expression  for  the  total  flux  is 
obtained  in  the  following  manner: 


,     . 
(15) 

which  may  be  written  in  the  form 

(16) 


The  numerator  of  the  right-hand  member  is  of  the  nature  of  a 
driving  force,  the  denominator,  an  opposition  factor  depending 
upon  the  medium,  and  their  ratio,  the  effect  produced.  This 


MAGNETISM  101 

equation  is  called  the  "Law  of  the  Magnetic  Circuit"  and  is 
written 

..     -r,.  Magnetomotive  Force 

Magnetic  Flux  =  - 

Reluctance 

83.  Magnetic    flux,    which   represents    the    total   number   of 
lines  of  induction,  is  analogous  to  flow  of  heat  in  calorimetery, 
and  to  current  in  electricity.     Jt  forms  a  closed  path  which  may 
be  spread  out  over  a  targe  area  in  some  places  and  be  concen- 
trated within  narrow  limits  in  others.     The   unit  of  magnetic 
flux  is  called  the  maxwell  and  is  represented  by  one  magnetic 
line  of  induction  through  the  total  cross  sectional  area  of  the 
magnetic  circuit.     Thus,  if  in  a  magnetic  circuit,  there  are  one 
thousand  lines,  the  flux  is  said  to  be  one  thousand  maxwells. 
In  engineering  practice,  it  is  customary  to  define  flux  on  the 
basis  of  the  E.M.F.  induced  in  a  conductor  which  cuts  it. 

Definition. — //,  in  a  moving  conductor,  the  induced  E.M.F.  is  one 
electromagnetic  unit,  the  flux  cut  per  second  is  one  maxwell. l 

84.  The   magnetic    induction    is    defined    as    the   total   flux 
divided  by  the  area,  and  is,  accordingly,  the  flux  density.     The 
unit    of    magnetic    induction    is    the    gauss. 

Definition. — Unit  induction,  or  one  gauss,  exists  in  a  magnetic 
circuit  in  which  the  flux  density  is  one  maxwell  per  square  centi- 
meter. Thus 

~  Maxwells 

Gausses  =  ~ ^     ,. — 

Square  Centimeters 

85.  Magnetomotive  force  may  be  regared  as   the   cause  of 
magnetic  flux.     It  is  analogous  to  electromotive  force  in  the 
electric  circuit  and  is  measured  in  a  similar  manner.     Just  as  the 
electromotive  force  of  an  electrical  circuit  is  the  work  required 
to  carry  unit  electrical  charge  once  around  the  circuit,  so  the 
magneto-motive  force  in  a  magnetic  circuit  is  the  work  required 
to  carry  unit  magnetic  pole  once  around  the  circuit.     The  unit 
of  magnetomotive  force  is  the  gilbert. 

Definition. — //  the  work  required  to  carry  a  unit  magnetic  pole 
once  around  a  magnetic  circuit  is  one  erg,  the  magnetomotive  force 
is  one  gilbert. 

In  case  the  magnetomotive  force  is  produced  by  a  current  in  a 
closed  solenoid,  as  in  the  above  illustration,  its  value,  as  given 

1  Note. — The  Units  for  the  quantities  involved  in  the  magnetic  circuit 
here  described  were  adopted  by  the  International  Electrical  Congress  at 
Paris,  in  1900. 


102  ELECTRICITY  AND  MAGNETISM 

by  equation  (16)  is  .47rNI.  The  product  NI  is  called  the  ampere 
turn,  and  differs  from  magnetomotive  force  only  by  the  constant 
factor  ATT  =  1.26.  Thus 

M.M.F.  in  Gilberts  =  ATT  Ampere  Turns. 

Magnetomotive  force,  being  thus  measured  in  terms  of  work 
per  unit  pole,  is  difference  of  magnetic  potential.  Accordingly, 
if  H  is  the  average  value  of  the  magnetic  field  strength  between 
two  equipotential  surfaces,  s  cms.  apart,  having  magnetic 
potentials  MI  and  Mz,  respectively, 

<»> 


where  AM  and  As  represent  small  differences  in  M  and  s,  respec- 
tively. Allowing  the  equipotential  surfaces  to  approach  indefi- 
nitely close  to  one  another,  the  limiting  value  of  this  ratio  gives 
the  actual  field  strength  at  a  given  point.  Thus 


Magnetic  field  strength  is  the  change  in  magnetic  potential  per 
centimeter  in  the  direction  of  H  or  the  magnetic  potential 
gradient.  The  unit  of  magnetic  field  strength  is  called  the 
gilbert  per  centimeter. 

86.  Reluctance  is  the  resistance  a  body  offers  to  being  mag- 
netized and  depends  upon  the  constants  of  the  circuit  in  a  manner 
similar  to  resistance  in  the  electrical  circuit.  As  seen  from 
eq.  (16),  it  is  directly  proportional  to  the  length  and  in- 
versely proportional  to  the  area  and  the  permeability  of  the 
medium.  Permeability  thus  corresponds  to  specific  conductivity, 
and  its  reciprocal,  cerresponding  to  specific  resistance  or 
resistivity,  is  often  called  "  reluctivity."  The  unit  of  reluctance 
is  defined  in  terms  of  the  law  of  the  magnetic  circuit  and  is 
called  the  oersted. 

Definition.  —  //,  in  a  magnetic  circuit,  the  flux  is  one  maxwell 
when  the  magnetomotive  force  is  one  gilbert,  the  reluctance  is  one 
oersted. 

Reluctances,  like  resistances,  may  be  joined  in  series  or 
parallel  to  form  complex  circuits,  and  laws  similar  to  those  for 
resistances  hold. 

1.  For  reluctances  joined  in  series,  the  total  reluctance  is  the 
sum  of  the  individual  reluctances. 


MAGNETISM  103 

2.  For  reluctances  joined  in  parallel,  the  reciprocal  of  the 
total  reluctance  is  the  sum  of  the  reciprocals  of  the  individual 
reluctances. 

87.  Limitations. — While  the  idea  of  the  magnetic  circuit  is  an 
extremely  useful  one  for  purposes  of  calculation,  it  must  not  be 
regarded  as  a  true  physical  concept,  such  as  the  electrical  circuit, 
but  merely  as  an  analogy  serving  a  useful  purpose.     Among  the 
respects  in  which  the  analogy  fails  are  the  following : 

1.  There  is  no  such  thing  as  a  magnetic  substance  in  the  sense 
in  which  we  have  used  it,  and  hence  there  can  be  no  magnetic 
flow. 

2.  When  once  the  magnetic  flux  has  been  established,    no 
energy  is  required  to  maintain  it,  and  there  is  nothing  correspond- 
ing to  the  I2R  consumption  of  energy  in  the  electric  circuit. 

3.  The  reluctance  of  a  circuit  containing  ferro-magnetic  mate- 
rial is  not  a  constant  for  a  given  set  of  physical  conditions  but 
varies  with  the  flux,  while  the  resistance  of  an  electric  circuit 
is  independent  of  the  current  flowing. 

4.  For  ferro-magnetic  materials,  the  reluctance  is  not  a  single 
valued  function  of  the  flux,  but  depends  upon  the  magneto-motive 
forces  to  which  they  previously  have  been  subjected.     In  other 
words,  there  is  no  analogy,  in  the  electric  circuit,  to  Hysteresis. 

88.  Magnetization  Curves. — Para-and  diamagnetic  substances 
are  characterized  by  the  fact  that,  under  a  given  set  of  physical 
conditions,  the  permeability  remains  constant;  that  is,  as  the 
magnetizing  field  is  changed,  the  induction  changes  by  propor- 
tional amounts.     This,  however,  is  not  true  of  ferro-magnetic 
substances.     If  a  piece  of  unmagnetized  iron,  for  example,  is 
placed  in  a  field  which  may  be  varied  at  will,  it  is  found,  starting 
with  H  =  0  and  gradually  increasing  it,  that  the  induction  B 
increases  slowly  at  first,  remaining  nearly  proportional  to  the 
field;  then  increases  rapidly,  for  a  certain  interval  of  H,  after 
which  a  further  increase  produces  only  relatively  small  increments 
in  B.  The  curve  showing  the  values  of  induction  for  different 
magnetizing  fields  is  called  the  "  magnetization  curve,"  and  is 
represented  by  OB  of  Fig.  58.     The  three  parts  of  the  curve, 
differentiated  by  rather  abrupt  changes  in  slope,  are  accounted 
for    by    assuming    that,    in    the    unmagnetized   condition,    the 
magnetic    axes    of    the    molecular    magnets    are    distributed 
entirely  at  random,  as  many  pointing  in  one  direction  as  in  any 
other;  and  that  the  magnetic  circuits,  of  which  they  form  parts, 


104 


ELECTRICITY  AND  MAGNETISM 


are  small  closed  curves.  Under  the  action  of  a  weak  magnetic 
field,  these  molecular  magnets  are  all  sprung  to  a  slight  extent 
from  their  initial  positions,  giving  a  resultant  component  in  the 
direction  of  the  applied  field,  the  amount  of  deformation  being 
proportional  to  the  field.  Thus  the  part  of  the  curve  near  the 
origin  is  obtained.  With  a  further  increase  of  field,  some  of 
these  local  magnetic  circuits  are  broken,  and  new  alignments 
formed;  giving  chains  of  molecules  of  considerable  length.  As 
each  local  circuit  breaks,  becoming  part  of  a  chain,  neighboring 


FIG.  58. — Magnetization  and  hysteresis  curves. 

groups  become  unstable,  break,  and  form  other  chains,  thus 
giving  a  sort  of  spontaneous  magnetization,  resulting  in  changes 
in  induction  much  greater  than  required  for  proportionality  to 
changes  in  field.  Thus  the  steep  part  of  the  curve  is  given.  As 
the  condition  is  approached  in  which  all  the  local  groups  have 
been  broken  up  and  the  molecules  placed  in  complete  alignment, 
the  iron  is  said  to  become  saturated,  and  further  increases  in  field 
produce  only  small  changes  in  induction.  So  the  upper  part  of 
the  curve  which  is  nearly  horizontal  is  obtained. 

89.  Hysteresis. — If,  after  the  induction  has  been  carried  to 
the  point  marked  +  Bmax  on  the  curve  of  Fig.  58,  the  magnetizing 
field  is  gradually  reduced,  the  induction  does  not  retrace  the 
magnetization  curve,  but  takes  on  values,  for  a  given  field,  greater 
than  those  for  the  magnetization  curve;  and  when  H  has  been 
reduced  to  zero,  an  amount  of  induction  indicated  by  BR  still 
persists.  If  a  reverse  field  is  applied,  the  induction  rapidly  falls; 
and  when  a  certain  value,  —  Hc,  has  been  reached,  the  resultant 
induction  is  zero;  after  this,  a  further  negative  increase  in  field  to 


MAGNETISM  105 

—  #max  gives  a  reversed  value  of  induction  —Bmax  equal  in  magni- 
tude to  +Bmax.  With  a  gradual  increase  in  H  to  its  original 
positive  value,  B  assumes  values  shown  by  the  lower  curve  of  the 
figure,  symmetrical  with  respect  to  the  origin  with  the  upper  one 
just  described.  This  tendency  of  any  material  to  persist  in  a 
given  magnetic  state  is  kno»wn  as  "  hysteresis,"  and  the  corre- 
sponding curve  is  called  the  hysteresis  curve.  BR  is  called  the 
retentivity,  and  Hc,  tfie  coercive  field. 

It  may  be  shown  that  the  area  of  the  hysteresis  loop  is  a 
measure  of  the  energy  consumed  by  molecular  friction  in  each 
cubic  centimeter  of  material  when  carried  once  through  a  mag- 
netic cycle.  For  this  purpose,  let  us  refer  to  the  ring  specimen 
described  in  Art.  75,  and  use  the  nomenclature  there  indicated. 
The  method  of  proof  is  based  upon  the  fact  that,  as  the  current  in 
the  magnetizing  coil  is  changed,  producing  changes  of  flux  in  the 
ring,  a  counter  E.M.F.  is  induced,  against  which  the  magnetizing 
current  must  flow.  The  electrical  energy  which  thus  disappears 
is  the  energy  consumed  by  hysteresis  and  reappears  in  the  form  of 
heat  within  the  ring.  Let  i  represent  the  instantaneous  magnetiz- 
ing current  and  let  dB  and  d(f>  be  the  changes  in  induction  and 
flux,  respectively,  when  a  change  di  occurs  in  the  magnetizing 
current.  If  dt  represents  the  time  required  for  this  change  to 
take  place,  the  energy  dw  consumed  during  the  change  is  given  by 

dw  =  eidt  (19) 

But 


Therefore 

dw  =  NAidB  (21) 

Since 

„  _4orNi 
~^T> 
we  have 

,  M  =  g-  (22) 

Substituting 


where  V  is  the  volume  of  the  ring.     Summing  up  for  the  complete 
cycle,  we  have 

Cdw  =  W  =  ^  (HdB  =  ¥-  (area  of  loop)  (24) 


106  ELECTRICITY  AND  MAGNETISM 

It  is  thus  seen  that  the  area  of  the  loop  divided  by  4?r  gives  the 
energy  lost  per  cycle  per  cubic  centimeter  of  material.  The 
shape  of  the  loop  varies  with  the  quality  of  the  iron;  hard  steels 
have  both  a  high  retentivity  and  coercive  force;  soft  steels,  a 
high  retentivity  but  a  low  coercive  force;  while  Swedish  iron  has 
both  a  low  retentivity  and  low  coercive  force.  For  a  given  speci- 
men, the  area  of  the  loop  depends  upon  the  limits  of  induction. 
Steinmez  has  made  an  exhaustive  study  of  this  relation  and  has 
found  that  the  energy  lost  is  proportional  to  the  1.6  power  of  the 
maximum  induction.  Expressed  in  symbols, 

W  =  KB1-*  (25) 

K  is  called  the  Steinmetz  coefficient. 

90.  Practical  Methods.1 — For  the  measurement  of  magnetic 
induction,  there  are  three  general  methods,  each  of  which  possess 
certain  advantages  as  well  as  disadvantages.  They  may  be 
classified  as  follows: 

1.  The  Traction  Method. 

2.  The  Magnetometer  Method. 

3.  The  Ballistic  Method. 

The  first  method  consists  in  measuring  the  mechanical  force 
required  to  pull  the  magnetized  specimen  away  from  a  massive 
piece  of  iron.  Since  the  specimen  induces  in  the  block  at  the  point 
of  contact  a  pole  of  strength  equal  and  opposite  to  its  own,  the 
force  required  to  separate  them  is  proportional  to  the  square  of 
the  intensity  of  magnetization.  In  the  second  method,  the 
specimen  is  made  in  the  form  of  a  rod  or  elongated  ellipsoid  and 
magnetized  by  being  placed  within  a  long  solenoid.  Its  mag- 
netic moment  is  determined  by  observing  the  deflection  it 
produces  upon  a  small  compass  needle,  called  a  magnetometer, 
placed  near  it.  From  the  magnetic  moment,  the  intensity  of 
magnetization,  and  hence  the  induction,  may  be  computed. 
In  the  ballistic  method,  the  specimen  under  test  forms  the  whole 
or  part  of  a  closed  magnetic  circuit,  wound  with  suitable  mag- 
netizing coils,  and  also  a  secondary  coil,  connected  to  a  ballistic 
galvanometer.  Any  change  in  flux  induces  in  the  secondary  a 
quantity  of  electricity  which  is  measured  by  the  ballistic  gal- 
vanometer and  from  this  quantity  the  change  in  flux  is  computed. 
From  the  standpoint  of  accuracy  and  ease  of  performance  the 

1  EWING,  Magnetic  Induction  in  Iron,  chap.  II. 
DuBois,  The  Magnetic  Circuit,  chap.  XI. 


MAGNETISM 


107 


ballistic  method  is  much  to  be  preferred  and  is  the  only  one  which 
will  be  considered  here.  • 

91.  Hopkinson's  Bar  and  Yoke.1 — This  is  an  application  of  the 
ballistic  method  in  which  the  samples  to  be  tested  are  in  the 
form  of  rods,  closely  fitted  into  holes  in  a  heavy  yoke  of  soft  iron. 
The  arrangement  is  shown  in;  Fig.  59,  where  YY  is  the  yoke  and 
CC'  the  specimen  under  test.^  MM  are  the  magnetizing  coils 
and  F  the  secondar/  coil.  The  magnetic  lines  through  the 
specimen  return,  half  through  the  upper  and  half  through  the 
lower  part  of  the  yoke.  Since  the  cross  section  of  the  yoke  is 
large  in  comparison  with  that  of  the  specimen,  its  reluctance 


M 


FIG.  59. — Hopkinson's  bar  and  yoke. 


may  be  neglected  without  appreciable  error  and  the  entire 
reluctance  be  considered  as  that  part  of  the  bar  within  the  slot. 
The  rod  consists  of  two  parts  joined  at  a  point  a  little  to  the 
right  of  F,  one  of  which  is  clamped  at  C',  while  the  other  may  be 
drawn  out  by  the  ring  at  C.  Springs  are  attached  to  the  second- 
ary coil  F,  generally  called  the  "flip"  coil,  which  runs  between 
guides.  While  the  bar  is  being  subjected  to  the  desired  mag- 
netizing field,  the  part  C  is  quickly  withdrawn,  releasing  F  which 
is  jerked  suddenly  out  of  the  field,  cutting  the  entire  flux  through 
the  specimen.  The  induction  B,  for  a  given  value  of  H,  is 
computed  in  the  following  manner: 


1  EWING,  Magnetic  Induction  in  Iron,  p.  67-92. 
SMITH,  Electrical  Measurements,  chap.  X. 


108  ELECTRICITY  AND  MAGNETISM 

Let  K  =  constant  of  the  ballistic  galvanometer 

Uf  —  turns  on  flip  coil 
R  =  total  resistance  of  secondary  circuit 
i  =  instantaneous  current  in  secondary  circuit 
e  =  instantaneous  E.M.F.  induced  in  secondary  circuit 

df  =  deflection  of  galvanometer 
4>  =  total  flux  in  specimen 

As  =  area  of  specimen 

Nm  =  magnetizing  turns 

Im  =  magnetizing  current 

LR  =  length  of  rod  (length  of  slot  in  yoke) 

The  quantity  Q  of  electricity,  expressed  in  coulombs,  discharged 
through  the  galvanometer  is  given  by  the  expression 

Q  =  Kdf  =  fidt  (26) 

But 


R       W*R  dt  ="  108#  dt 
Hence 

Therefore 


The  constant  K  of  the  ballistic  galvanometer  may  be  obtained  by 
means  of  the  standard  solenoid  described  in  Art.  26.  Substitut- 
ing the  value  of  K  from  Eq.  36. 


I 

-D    =    -777  -  T-'-'O/.  (60) 


The  field  strength  to  which  the  specimen  is  subjected  is  given  by 
the  regular  formula  for  the  ring  solenoid 

4-TrN     T 
TT  ^TT2\  mJ-  m  /Oi\ 

" 


92.  Experiment  14.  Magnetization  Curves  by  Hopkinson's 
Bar  and  Yoke.  —  Connect  the  apparatus  as  shown  in  Fig.  60, 
where  C  and  Y  are  the  bar  and  yoke,  respectively.  G  is  a  ballistic 
galvanometer  and  DE  a  standard  solenoid  for  calibrating  it. 
Since  a  considerable  range  of  currents  will  be  required,  use  two 
ammeters,  one  of  range  0-15  amperes  and  the  other,  a  millam- 
meter  connected  as  shown,  where  Si  is  a  knife  switch  which  should 
be  left  closed  during  all  manipulations.  Open  Si  when  it  is 


MAGNETISM 


109 


desired  to  read  the  millammeter  and  then  only  when  the  0-15 
ammeter  indicates  a  current  less  than  the  full  scale  reading  of 
the  millammeter.  S2  is  also  a  knife  switch.  First  compute,  by 
means  of  eq.  (31),  the  upper  and  lower  limits  of  current  required 
for  field  strengths  ranging  from  1  to  120  gilberts  per  centimeter. 
Before  proceeding  to  test  a  specimen,  it  must  first  be 


B 


l VNAA^WVVA 

R 


FIG.  60. — Connections  for  Hopkinson's  bar  and  yoke. 


demagnetized.  This  is  done  by  applying  a  magnetizing  current, 
somewhat  greater  than  that  required  for  the  maximum  test  field, 
and  reducing  it  by  small  steps,  reversing  the  commutator  at 
each  step  until  a  current  barely  readable  on  the  millammeter  has 
been  reached.  The  rate  of  commutation  should  not  exceed  20 
reversals  per  minute.  In  this  way,  the  specimen  is  magnetized 
first  in  one  direction,  and  then  in  the  other,  each  time  to  a  less 


110  ELECTRICITY  AND  MAGNETISM 

extent,  until  finally  all  magnetism  has  disappeared.  This 
should  be  done  with  the  flip  coil  out  of  position  or  with  the  second- 
ary circuit  broken,  to  avoid  damaging  the  galvanometer.  To 
test  for  residual  magnetism,  flip  the  coil  with  no  current  in  the 
magnetizing  coils.  A  deflection  not  exceeding  a  millimeter 
should  be  obtained.  Next,  determine  the  constant  of  the  gal- 
vanometer. To  do  this,  set  the  double  pole  double  throw  switch 
so  as  to  connect  in  circuit  the  primary  of  the  standard  solenoid, 
and,  with  a  steady  current  of  about  two  amperes  flowing,  reverse 
the  commutator  in  such  a  direction  as  to  cause  the  galvanometer 
to  swing  to  high  numbers.  Make  several  determinations  in  this 
manner,  using  such  values  of  primary  current  as  will  give  deflec- 
tions ranging  from  2  to  14  centimeters.  It  is  necessary  here  to 
reverse  the  primary  current,  not  simply  to  make  or  break  it, 
since  that  is  the  assumption  on  which  the  formula  for  the  deter- 
mination of  the  constant  was  derived.  Use  the  average  value 
of  the  ratio  of  current  to  deflection  in  eq.  (30). 

Everything  is  now  ready  for  the  test  proper.  Set  the  double 
pole  double  throw  switch  again  so  as  to  include  the  magnetizing 
coils  and  the  rheostat  so  as  to  include  the  maximum  resistance. 
Close  the  battery  circuit  and  bring  the  current  up  to  the 
smallest  value  computed  above.  Flip  the  coil  and  note  the 
deflection  of  the  galvanometer,  which  should  swing  in  the  same 
direction  as  used  when  determining  its  constant.  Obtain,  in  this 
manner,  about  fifteen  points  on  the  magnetization  curve,  spaced 
more  closely  together  in  the  lower  part  of  the  field  strength  range, 
where  the  curve  rises  steeply.  Caution. — Points  must  be  taken 
always  with  increasing  field  strength.  Do  not  allow  the  current 
to  rise  too  high  and  then  decrease  it.  Obtain  data  for  the  magne- 
tization curves  for  the  samples  of  iron  furnished.  Check  your 
galvanometer  constant  before  and  after  taking  each  set. 

Report. — 1.  Plot  magnetization  curves  for  the  four  samples 
using  B  as  ordinates  and  H  as  abscissas. 

2.  Calculate  the  permeability  for  each  value  of  H  and,  on  a 
separate  sheet,  plot  permeability  as  ordinates  and  field  strength 
as  abscissas  for  each  sample. 

3.  For  the  maximum  field  strength,  compute  the  magneto- 
motive force,  total  flux  and  reluctance  of  the  magnetic  circuit  for 
each  sample,  expressing  each  quantity  in  its  proper  units.     What 
is  the  relation  between  maxwells  and  gausses? 


MAGNETISM 


111 


93.  The  Rowland  Ring.1 — In  the  bar  and  yoke  method  des- 
cribed above,  errors  are  introduced  due  to  imperfect  magnetic 
contact  between  the  ends  of  the  rods  and  the  rod  and  yoke.  This 
objection  is  overcome  by  making  the  specimen  in  the  form  of  a 
ring,  either  turned  true  in  the  lathe  from  a  solid  block,  or  built  up 
of  sheet  stampings.  The  magnetizing  coil  is  then  wound  uni- 


/VWVWW 
R 


FIG.  61. — Connections  for  Rowland  Ring  Method. 

f ormly  over  the  entire  magnetic  circuit  with  the  secondary  wound 
over  the  primary.  Since  the  turns  on  the  inner  side  of  the  ring 
are  closer  together  than  on  the  outer,  the  former  part  of  the  ring 
will  be  subject  to  a  greater  mange tizing  force  than  the  latter,  and 
therefore,  the  thickness  of  the  ring  should  be  small  compared  to  its 

1  EWING.  Magnetic  Induction  in  Iron,  chap.  III. 
SMITH,  Electrical  Measurements,  chap.  XII. 
ROWLAND,  Phil.  Mag.  vol.  46,  1873,  p.  151. 


112  ELECTRICITY  AND  MAGNETISM 

diameter,  the  requisite  area  being  obtained  by  increasing  the 
height.  As  the  magnetic  circuit  cannot  be  broken,  it  is  impossi- 
ble to  obtain  any  measurement  of  the  magnetic  state  of  a  given 
ring,  so  the  method  of  observation  is  limited  to  measurements  of 
changes  of  magnetic  state  produced  by  definite  changes  in 
magnetizing  force. 

A  magnetization  curve  may  be  obtained  by  a  series  of  reversals 
carried  out  in  the  following  manner :  Suppose  the  iron  to  be  in  an 
unmagnetized  condition.  Apply  a  weak  magnetizing  field.  The 
induction  rises  a  small  amount  along  the  desired  magnetization 
curve.  Reverse  the  magnetizing  field.  This  causes  the  induc- 
tion to  change  along  the  upper  half  of  a  small  hysteresis  cycle, 
taking  on  a  value  the  negative  of  what  it  had  before  the  reversal 
occurred.  The  change  of  induction,  which  is  measured  by  the 
ballistic  galvanometer,  is  then  twice  the  total  induction  existing 
before  the  reversal.  Now  increase  the  field  to  a  somewhat  larger 
value,  thus  carrying  the  induction  to  a  higher  point  on  the  curve. 
By  again  measuring  the  change  of  induction  on  reversal,  twice  the 
new  value  of  induction  is  obtained,  and  so  on  for  a  series  of  points 
determining  the  entire  magnetization  curve.  By  this  process, 
the  iron  is  taken  around  a  series  of  successively  larger  and  larger 
hysteresis  cycles,  the  apexes  of  whose  corresponding  curves  lie 
upon  the  desired  magnetization  curve.  The  induction  B,  for  a 
given  value  of  H,  is  computed  in  the  following  manner: 

Let  K  =  constant  of  the  ballistic  galvanometer 
Ns  =  turns  on  secondary  coil 
R  =  total  resistance  of  secondary  circuit 
e  =  instantaneous  E.M.F.  in  secondary  circuit 
i  =  instantaneous  current  in  secondary  circuit 
ds  =  deflection  of  galvanometer 
<t>  =  total  flux  in  ring 
AR  =  cross  sectional  area  of  ring 
Nm  =  magnetizing  current 
LR  =  mean  circumference  of  ring 

The  quantity  of  electricity  Q,  expressed  in  coulombs,  discharged 
through  the  galvanometer  is  given  by  the  expression 

Q  =  Kds  =  fidt  (32) 

But 

.      _e        Nsdcf>       N8AR  dB  ,    . 

~  R    ~  IWRdt        W*R    dt 


MAGNETISM  113 

Substituting 


'          •  -  »> 

The  constant  K  of  the  ballistiq  galvanometer  may  be  obtained  by 
means  of  the  standard  solenoid  inethod,  as  described  in  Art.  26. 
Substituting  the  value  T>f  K  from  eq.  (36),  we  have 

4irNnA    I 
''  ds 


The  field  strength  to  which  the  specimen  is  subjected  is  given  by 
the  formula 


H  =  (37) 


94.  Experiment  15.  Magnetization  Curves  by  the  Rowland 
Ring  Method.  —  Connect  the  apparatus  as  shown  in  Fig.  61. 
R  is  the  ring  specimen  under  test,  and  Nm  and  Na  the  primary  and 
secondary  windings,  respectively.  G  is  a  ballistic  galvanometer, 
and  DE  a  standard  solenoid  for  calibrating  it  Since  a  consider- 
able range  of  current  will  be  required,  use  two  ammeters,  one  of 
range  0-15  amperes  and  the  other  a  millammeter,  connected  as 
shown,  where  Si  is  a  knife  switch  which  should  be  left  closed 
during  all  manipulations.  Open  Si  when  it  is  desired  to  read  the 
millammeter  and  then  only  when  the  0-15  ammeter  indicates  a 
current  less  than  the  full  scale  reading  of  the  millammeter.  First, 
compute  by  means  of  eq.  (37),  the  upper  and  lower  fields  from  .5 
to  100  gilberts  per  centimeter. 

Before  proceeding  to  test  a  specimen,  it  must  first  be  demagne- 
tized. This  is  done  by  applying  a  magnetizing  current  somewhat 
greater  than  that  required  for  the  maximum  test  field,  and  reducing 
it  by  small  steps,  reversing  the  commutator  at  each  step  until  a 
current  barely  readable  on  the  millammeter  has  been  reached. 
The  rate  of  commutation  should  not  exceed  20  reversals  per  minute. 
This  should  be  done  with  the  secondary  circuit  broken  to  avoid 
damaging  the  galvanometer.  Next  determine  the  constant  of  the 
galvanometer.  To  do  this,  set  the  double  pole  double  throw 
switch  so  as  to  connect  in  circuit  the  primary  of  the  standard  sole- 
noid and  with  a  steady  current  of  about  2  amperes,  reverse  the 
commutator  in  such  a  direction  as  to  cause  the  galvanometer  to 
swing  to  high  numbers.  Make  several  determinations  in  this 


114  ELECTRICITY  AND  MAGNETISM 

manner,  using  such  values  of  current  as  will  give  galvanometer 
deflections  ranging  from  2  to  14  centimeters.  It  is  necessary  here 
to  reverse  the  primary  current,  not  simply  to  make  or  break  it, 
since  that  is  the  assumption  on  which  the  formula  for  the  galvano- 
meter constant  was  derived.  Use  the  average  value  of  the  ratio 
of  current  to  deflection  in  eq.  (36). 

Everything  is  now  ready  for  the  test  proper.  Set  the  double 
pole  double  throw  switch  again  so  as  to  include  the  primary  on 
the  ring,  and  the  rheostat  R  so  as  to  include  the  maximum  resis- 
tance. Close  the  battery  circuit  and  bring  the  current  up  to  the 
smallest  value  computed  above.  Bring  the  galvanometer  to 
rest,  reverse  the  primary  current,  and  note  the  galvanometer 
deflection.  Now  bring  the  commutator  back  to  its  original  posi- 
tion, increase  the  current  to  a  slightly  greater  value,  and  read  the 
galvanometer  deflection  again  on  reversal.  Obtain,  in  this  man- 
ner, about  15  points  on  the  magnetization  curve,  spaced  more 
closely  together  on  the  lower  part  of  the  field  strength  range 
where  the  rise  is  rapid.  Caution. — Succeeding  points  must 
always  be  taken  with  increasing  field  strength.  Do  not  allow 
the  current  to  rise  too  high  and  then  decrease  it.  Obtain  data 
for  the  magnetization  curves  for  two  samples  of  iron.  Check 
your  galvanometer  constant  before  and  after  taking  each  set. 

Report. — 1.  Plot  magnetization  curves  for  the  two  samples, 
using  B  as  ordinates  and  H  as  abscissas. 

2.  Compute  the  permeability  for  each  value  of  //,  and,  on  a 
separate  sheet,  plot  permeabilities  as  ordinates  and  field  strengths 
as  abscissas. 

3.  For  the  maximum  field  strength,  compute  the  magneto- 
motive force,  total  flux,  and  reluctance  of  the  magnetic  circuit  for 
each  sample,  expressing  each  quantity  in  its  proper  units.     What 
is  the  relation  between  maxwells  and  gausses? 

95.  Experiment  16.  Hysteresis  Curves  by  the  Rowland  Ring 
Method.1 — Connect  the  apparatus  as  indicated  in  Fig.  61,  and 
observe  the  precautions  regarding  use  of  ammeters,  switches, 
rheostats,  etc.,  indicated  in  Exp.  15.  Instead  of  starting  with 
zero  field  and  making  changes  of  induction  which  are  sym- 
metrical with  respect  to  the  origin,  as  in  the  case  of  the  magneti- 
zation curve  by  reversals,  start  here  with  the  maximum  field  and 
make  changes  of  induction  by  passing  first  to  the  retentivity 

1  EWING,  Magnetic  Induction  in  Iron,  chap.  V. 
TAYLOR,  Physical  Review,  vol.  23,  p.  95. 


MAGNETISM  115 

point  and  then  away  from  it.  All  measurements  of  induction  are 
accordingly  to  be  made  with  respect  to  the  retentivity  point, 
and  we  will,  for  the  moment,  regard  this  point  as  the  origin  from 
which  the  upper  half  of  the  hysteresis  curve  is  to  be  plotted. 
The  method  will  be  made  clear  by  reference  to  Fig.  58.  Apply 
first  the  maximum  field,  giving  -f  J5max  on  the  curve.  Now 
reduce  the  field  to  zero.  The  induction  changes  along  the  upper 
part  of  the  curve,  and  "goes  to  the  retentivity  point,  the  actual 
change  being  equal  to  BI.  Now  apply  the  field  —  //max-  The 
induction  changes  along  the  curve  from  BR  to  Bm&x,  the  actual 
change  in  induction  being  B2.  BI  and  B2  are  thus  located  on  the 
curve  with  BR  as  the  origin.  An  intermediate  point,  such  as 
Bs  may  be  obtained  by  applying  again  the  field  -\-Hmax  and  slowly 
reducing  to  +//3  without  breaking  the  magnetizing  current. 
If  the  magnetizing  current  is  now  broken,  the  induction  again 
returns  to  BR  and  the  change,  which  is  measured  by  the  galva- 
nometer deflection,  is  B3.  This  locates  B$  with  respect  to  BR. 
The  corresponding  point,  B4  may  be  obtained  by  applying  the 
field  —  Hz  and  observing  the  throw  of  the  galvanometer.  In  a 
similar  manner,  a  series  of  points,  corresponding  to  pairs  of 
positive  and  negative  values  of  //,  may  be  obtained  and  the 
upper  half  of  the  curve  plotted  with  respect  to  BR. 

The  actual  manipulation  of  switches  is  as  follows:  Obtain  the 
constant  of  the  galvanometer  as  explained  in  Exp.  14.  With 
the  galvanometer  circuit  broken,  set  the  rheostat  to  give  the 
maximum  magnetizing  current.  Reverse  this  current  several 
times  through  the  primary  coil  of  the  ring  to  remove  the  effects 
of  previous  magnetization,  and  thus  make  sure  that  the  iron  will 
follow  the  cycle  desired.  With  maximum  current  flowing,  close 
the  secondary  circuit  and  bring  the  galvanometer  to  rest. 
Break  the  primary  circuit  by  the  switch  $2  and  observe  the  throw 
of  the  galvanometer  which  measures  BI.  Bring  the  galvanometer 
again  to  rest  with  the  secondary  circuit  closed.  Reverse  the 
commutator.  Close  $2  and  note  the  deflection  of  the  galva- 
nometer which  measures  52.  Break  the  secondary  circuit,  reverse 
the  commutator,  bringing  the  induction  back  to  +#max-  Reduce 
the  current,  without  breaking  the  circuit,  to  give  a  value  +HS. 
Close  the  secondary  circuit  and  bring  the  galvanometer  to  rest. 
Break  the  primary  by  means  of  $2  and  the  throw  of  the  galva- 
nometer measures  53.  Bring  the  galvanometer  to  rest,  reverse 
the  commutator,  and  close  *S2.  The  deflection  of  the  galvanome- 


116  ELECTRICITY  AND  MAGNETISM 

ter  measures  £4,  the  induction  corresponding  to  —#3.  The 
other  points  on  the  curve  are  obtained  in  pairs  in  the  same  manner. 
It  is  important  to  notice  that  before  each  pair  of  observations 
is  taken  the  induction  must  first  be  returned  to  +#max,  otherwise 
a  different  cycle  will  be  carried  out  for  each  pair.  Obtain  in  this 
way  at  least  ten  pairs  of  values  for  B,  using  field  strengths  rang- 
ing from  .5  to  30  gilberts  per  centimeter.  It  will  assist  the 
calculation  if  deflections  corresponding  to  positive  and  negative 
fields  are  recorded  in  separate  columns.  Two  samples  are  to  be 
tested. 

The  calculation  of  the  values  of  B  is  carried  out  by  the  same 
formula  as  used  in  Exp.  14  except  here  we  wish  the  total  change 
in  induction  instead  of  half  of  it  as  was  the  case  there.  Accord- 
ingly the  limits  of  integration  in  eq.  (33)  are  0  and  B  instead  of 
-\-B  and  —  B,  giving  as  our  final  formula. 

SirNnA     I 
10LN.Ar     d    ds' 

Before  plotting  the  curve,  the  origin  should  be  changed  from  £R 
to  0.  This  is  accomplished  by  adding  BR  to  all  values  of  induc- 
tion corresponding  to  positive  fields  and  subtracting  all  values  of 
induction  corresponding  to  negative  fields  from  BR.  BR  is 
determined  from  the  relation 

BR  =  Bmax  -  B,  =  B*  +  B*  -  B,  (39) 

The  lower  half  of  the  curve,  being  symmetrical  with  the  upper,  is 
plotted  from  these  same  data,  merely  changing  the  signs  of  all 
values  of  B. 

Report. — 1.  Plot  the  hysteresis  curves  for  the  two  samples  of 
iron,  making  the  plots  as  large  as  convenient. 

2.  Measure  the  area  of  the  curves  by  means  of  a  planimeter,  and 
determine  the  energy  loss  per  cycle  per  cubic  centimeter.     Since 
it  is  not  convenient  to  plot  B  and  H  to  the  same  scale,  if  unit  length 
along  the  B  axis  represents  b  gausses,  and  unit  length  along  the 

H  axis,  h  gilberts  per  centimeter,  unit  area  will  represent  -r-  ergs 

47T 

per  cc. 

3.  Compute  the  Steinmetz  coefficient  for  each  sample. 


CHAPTER  IX 
SELF  AND  MUTUAL  INDUCTANCE1 

Hi 

96.  General  Principles.  —  Whenever  a  change  occurs  in  the 
number  of  magnetic  lines  linking  any  electrical  circuit,  there  is 
induced  within  the  circuit  an  electromotive  force,  which,  if  the 
circuit  is  closed,  will  cause  a  current  to  flow.  It  makes  no 
difference  by  what  means  this  change  is  produced;  whether  mag- 
nets in  the  neighborhood  are  moved,  currents  in  adjacent  circuits 
changed,  or  the  current  in  the  circuit  itself  varied,  the  nature  of 
the  induced  electromotive  force  is  the  same.  The  direction  of 
the  induced  electromotive  force  is  given  by  a  simple  rule  known  as 
Lenz's  law,  which  may  be  stated  as  follows:  Whenever  a  change 
occurs  in  an  electromagnetic  system,  the  direction  of  the  induced 
electromotive  force  is  such  that  the  magnetic  action  of  its  current 
opposes  the  change.  For  example,  if  the  north  pole  of  a  magnet 
is  moved  toward  a  closed  helix,  the  induced  current  flows  in  such 
a  direction  as  to  produce  a  north  pole  on  the  end  toward  the  mag- 
net, thus  tending  to  repel  it,  and  vice  versa,  when  it  is  with- 
drawn. The  magnitude  of  this  induced  E.M.F.  per  turn  is  given 
by  the  expression 


where  </>  is  the  total  flux  passing  through  the  turn  at  any  instant. 
If  the  change  of  flux  through  the  coil  is  produced,  not  by  mov- 
ing toward  it  a  magnetic  pole  but  by  changing  the  current 
in  another  coil  placed  near  it,  the  phenomenon  of  the  induced 
E.M.F.  is  called  mutual  induction.  The  coil  which  is  producing 
the  change  of  flux  is  called  the  primary  and  that  in  which  the 
E.M.F.  is  induced,  the  secondary.  If  the  current  in  the  primary 
of  two  coaxial  coils  is  rising,  let  us  say  in  the  clockwise  direction, 
on  looking  along  the  axis,  an  application  of  Lenz's  law  shows  that 
the  current  in  the  secondary  is  flowing  counter-clockwise,  while 
if  the  current  in  the  primary  is  decreasing,  the  secondary  current 

1  DUFF,  A  Textbook  of  Physics,  p.  445. 

REED  and  GUTHE,  College  Physics,  p.  365.     STARLING,  Electricity  and 
Magnetism,  chap.  XI. 

117 


118  ELECTRICITY  AND  MAGNETISM 

is  in  the  same  direction  as  the  primary.  Since  the  flux  through 
the  secondary  at  any  instant  is  proportional  to  the  current  in  the 
primary,  we  may  write  for  the  total  E.M.F.  in  the  secondary 

.  -  If  |  (2) 

where  i  is  the  primary  current  and  M  a  constant  depending  upon 
the  area  of  the  two  coils,  their  number  of  turns,  distance  apart, 
the  permeability  of  the  medium  surrounding  them,  etc.  M  is 
called  the  coefficient  of  mutual  inductance,  the  unit  of  which  has 
been  named  the  henry. 

Definition. — Two  coils  have  one  henry  of  mutual  inductance,  if, 
when  the  primary  current  is  changing  at  the  rate  of  one  ampere  per 
second,  the  induced  E.M.F.  in  the  secondary  is  one  volt. 

When  the  current  through  any  coil  is  changing,  there  is  a 
change  of  flux,  not  only  through  any  coil  in  the  neighborhood,  but 
also  through  the  coil  itself,  causing  an  induced  E.M.F.  within  it. 
This  phenomenon  is  known  as  Self  Induction.  The  direction  of 
this  E.M.F.,  considering  the  coil  to  be  its  own  secondary,  is 
determined  by  Lenz's  law,  as  given  above;  i.e.,  when  the  current 
is  rising,  the  induced  E.M.F.  is  in  such  a  direction  as  to  oppose 
the  current,  and  when  the  current  is  falling,  it  tends  to  maintain 
it.  The  induced  E.M.F.  always  opposes  any  change  in  the 
current  and  is  called  a  counter  E.M.F.  Since  the  flux  through 
the  coil  at  any  instant  is  proportional  to  the  current,  the  induced 
counter  E.M.F.  is  given  by 

•  -L*  ;..-.          .-.::    (3) 

where  i  is  the  current  at  any  instant  and  L  a  constant  depending 
upon  the  number  of  turns  in  the  coil,  its  area,  shape,  permeability 
of  the  surrounding  medium,  etc.  L  is  called  the  coefficient  of 
self  inductance  and  the  unit  is  the  henry. 

Definition. — A  coil  has  one  henry  of  self  inductance,  if,  when 
the  current  through  it  is  changing  at  the  rate  of  one  ampere  per 
second,  the  induced  counter  E.M.F.  is  one  volt. 

Since  the  henry  is  a  relatively  large  unit,  it  is  customary  in 
expressing  the  inductance  of  ordinary  coils,  to  use  a  unit  only 
one-thousandth  as  large,  called  the  millihenry.  Variable  stand- 
ards of  self  and  mutual  inductance  are  made  by  mounting 
two  coils  in  such  a  way  that  their  relative  positions,  and  hence 
their  inductive  interactions  may  be  changed.  If  the  coils  are 


SELF  AND  MUTUAL  INDUCTANCE 


119 


connected  in  circuit  separately,  one  being  used  as  the  primary  and 
the  other  as  the  secondary,  a  calibration  curve  may  be  made 
showing  the  mutual  inductance  between  them  for  various  posi- 
tions. If,  however,  they  are  connected  in  series  and  used  as  a 
single  coil,  a  variable  self  inductance  is  obtained,  since  the  resultant 
self  inductance  of  two  coils,  ^with  mutual  inductance  between 
them,  is  given  by  the  formula 


... 


L  =  Li  +  L2  ±  2M 


(4) 


where  LI  and  L2  are  the  separate  coefficients  of  self  inductance. 
If  the  coils  are  mounted  in  such  a  manner  that  advantage  may  be 
taken  of  both  positive  and 
negative  values  of  M,  variable 
self  inductances  of  consider- 
able range  may  be  obtained. 
Two  forms  of  variable  stand- 
ard are  in  common  use. 
Figure  62  represents  the 
Ayerton  and  Perry  variable 
inductor  which  consists  of 
two  coils  mounted  vertically 
one  of  which  is  fixed  and  the 
other  movable.  The  coils  are 
wound  on  spherical  surfaces, 
and  the  inner  one  rotates 
about  a  vertical  axis.  When 
the  planes  of  the  coils  are 
parallel,  the  resultant  self 
inductance  is  a  maximum  or  a 
minimunvaccording  as  the  mutual  is  positive  or  negative.  When 
the  coils  stand  at  right  angles  to  each  other,  the  resultant  self 
inductance  is  the  sum  of  the  self  inductances  of  the  two  coils, 
since  the  mutual  is  zero  for  this  position.  For  other  positions  of 
the  movable  coil,  intermediate  values  are  obtained.  The 
relation  between  resultant  self  inductance  and  angular  position 
is  nearly  linear.  Two  pointers  on  the  top  read,  one  the  angular 
position  of  the  coil  in  degrees,  the  other  the  self  inductance  in 
millihenries.  The  coils  are  joined  in  series  by  a  flexible  conductor. 
Separate  binding  posts  for  the  coils  are  usually  provided,  and,  when 
used  independently,  the  instrument  serves  as  a  variable  standard 
of  mutual  inductance  also. 


FIG.  62. — Ayerton  and  Perry  variable 
inductor. 


120  ELECTRICITY  AND  MAGNETISM 

The  other  instrument  is  known  as  the  Brook's  inductor  and  is 
illustrated  in  Fig.  63.  It  consists  of  six  coils  mounted  in  pairs  in 
three  hard  rubber  discs,  placed  one  above  the  other  in  a  hori- 
zontal position.  The  upper  and  lower  disks  are  fixed  and  the 
middle  one  rotates  between  them.  If  the  coils  are  joined  in  series 
and  connected  so  that  their  fields  on  one  side  are  all  directed 
upward,  and  on  the  other  side  downward,  the  resultant  self 
inductance  is  a  maximum;  but  if  the  middle  disk  is  turned  through 
180°,  the  mutual  inductance  between  the  fixed  and  movable 
coils  will  neutralize  the  self  inductance  and  the  resultant  will  be 


FIG.  63.  —  Brook's  variable  inductor. 

a  minimum.  By  properly  shaping  the  coils,  an  approximately 
linear  relation  is  obtained  between  angular  position  and 
inductance.  Separate  binding  posts  enable  the  coils  to  be  used 
independently  giving  also  a  variable  standard  of  mutual  induc- 
tance. The  instrument  is  provided  with  two  scales  which  read 
respectively  self-  and  mutual  inductance  in  millihenries. 

97.  Comparison  of  Inductances.1  —  Two  coefficients  of  self 
inductance  may  be  compared  by  a  bridge  method  in  which  the 
two  coils,  whose  inductances  are  to  be  compared,  form  two  arms 
of  the  ordinary  Wheatstone  bridge.  Let  LI  and  L2  of  Fig.  64  be 
two  inductances  having  resistances  R\  and  R%,  respectively, 
and  Rs  and  7?4  be  two  non-inductive  resistances,  and  let  the 
bridge  be  balanced  for  steady  currents,  as  explained  in  Art.  31, 
the  condition  for  which  is 


This  condition  signifies  that  when  the  currents,  ii  and  iz  are 
constant,  the  potentials  at  C  and  D  are  equal,  but  less  than  the 

1  CARHART  and  PATTERSON,  Electrical  Measurements,  p.  255. 
SMITH,  Electrical  Measurements,  p.  197-203. 
MAXWELL'S,  Elect,  and  Mag.,  vol.  2,  p.  367. 


SELF  AND  MUTUAL  INDUCTANCE 


121 


potential  at  A.  If  the  battery  key  KI  is  opened,  the  current 
ceases  to  flow  and  the  potentials  at  C  and  D  become  equal  to 
that  at  A.  When  KI  is  again  closed,  the  potentials  at  C  and  D 
on  account  of  the  counter  E.M.F.'s  of  self  induction  in  LI  and  L2 
will  not  necessarily  rise  at  the  same  rate,  although  they  will 
come  to  the  same  final  values.  Hence,  there  may  be  a  short 
interval  of  time  during  whiclu<a  difference  of  potential  exists 
between  C  and  D  giving  a  deflection  of  the  galvanometer  if  K2 
is  closed.  By  properly  adjusting  LI  and  L2  it  is  possible  to  cause 


FIG.  64. — Bridge  method  for  self-inductance. 

the  potentials  at  C  and  D  to  rise  at  the  same  rate  when  the  bridge 
is  balanced  for  both  steady  and  varying  currents.  The  condi- 
tions for  such  a  balance  is  obtained  in  the  ordinary  way,  except 
that  the  equations  must  include  terms  representing  the  fall  of 
potential  due  to  the  counter  E.M.F.  Equating  the  difference 
of  potential  at  any  instant  between  A  and  D  to  that  between  A 
and  C,  also  that  between  D  and  S,  to  that  between  C  and  S,  we 
have 

r>  •        T    dii       „  .        ,.    diz  ,f.N 

/»!»!  T  *J\     jj     =    •it  2*  2      I     •*--/2  ~TT  \d) 

at  at 

and 

T>    •  r>    •  //^\ 

xt/S^l    =    /t4'62  V.O/ 


122  ELECTRICITY  AND  MAGNETISM 

whence 


***1  Q          U-&2  ,  —  v 

dF  =  ft4dT  (7) 


Eliminating  t'2  and  -^,  we  have 


.  (8) 

Since  #i/?4  =  -R2/?3,  the  condition  for  steady  current  balance,  the 
condition  for  the  varying  current  or  inductive  balance  is 

Li/C4    =   L2/i3  (9) 

or 


.L/2          •t'4 

98.  Experiment  17.  Comparison  of  Two  Coefficients  of  Self 
Inductance  by  the  Bridge  Method.  —  Connect  the  apparatus  as 
shown  in  Fig.  64,  where  LI  is  the  unknown  inductance  and  L2 
a  variable  standard.  Rs  and  R^  may  be  two  ordinary  resistance 
boxes  with  non-inductively  wound  coils  connected  with  a  slide 
wire  LM  for  accurate  balancing.  K  i  and  K2  should  be  two 
ordinary  press  keys.  First,  using  for  B  a  battery  of  about  two 
volts,  obtain  a  steady  current  balance  by  closing  KI  first,  and  Kz 
after  the  current  has  had  time  to  rise  to  its  final  value.  Try  to 
keep  Rs  and  R^  between  one  hundred  and  three  hundred  ohms. 
For  the  inductive  balance,  use  a  battery  of  20  volts.  Close  K2 
first  a.nd  then  lightly  tap  K\,  never  leaving  it  closed  for  more  than 
an  instant,  since  the  large  currents  would  cause  too  great  a  heating 
of  the  resistances.  The  motion  of  the  galvanometer  in  this  case 
will  be  a  sudden  "kick"  not  a  steady  deflection.  Adjust  L2  until 
this  kick  has  disappeared.  Read  the  value  of  L2  and  compute  the 
value  of  LI  from  eq.  (10).  The  unknown  to  be  determined  con- 
sistsof  a  spool  with  two  independent  windings.  Determine  the 
inductance  of  each  separately,  then  join  them  in  series,  and 
determine  the  resultant  self  -inductances  with  their  mutual  induc- 
tances aiding  and  opposing,  making  in  all  four  measurements. 

Note.  —  It  may  happen  that  the  balance  point  lies  beyond  the 
range  of  the  variable  standard,  making  an  inductive  balance  im- 
possible. When  this  happens,  the  ratio  R%  to  R±  must  be  changed 
so  as  to  bring  the  balance  point  within  the  required  range. 
Since  a  steady  current  balance  must  always  be  obtained  first, 
this  requires  the  insertion  of  a  small  non-inductive  resistance  in 
series  with  either  R\  or  Rz  as  the  case  may  demand.  For  ex- 


SELF  AND  MUTUAL  INDUCTANCE 


123 


ample,  suppose  the  inductive  kick  of  the  galvanometer  decreases 
as  LI  is  increased  to  its  maximum,  but  cannot  be.  made  zero  or 
reversed.  The  combination  of  the  two  balance  conditions  gives 

Li    =    #3    =    #l 
i/2  R<1          RZ 

If,  then,  an  appropriate  resistance  is  connected  in  series  with  Ri 
the  new  steady  current  balance  Condition  will  give  a  larger  ratio 
of  Rs  to  7^4  thus  making'the  inductive  balance  possible.  If,  on  the 
other  hand,  L2  cannot  be  made  small  enough,  the  additional 
resistance  must  be  placed  in  series  with  7£2. 

Report. — 1.  Tabulate  your  data  for  the  determination  of  the 
four  inductances  as  indicated. 

2.  From  the  formula  L  =  LI  +  L2  +  2M,  compute  M  from 
the  cases  where  it  is  aiding  and  opposing  the  self  inductance. 
The  agreement  of  these  two  values  gives  a  check  on  the  accuracy 
of  your  work. 

3.  How  are  coils  wound  so  as  to  be  non-inductive? 


FIG.  65. — Mutual  inductance  by  Carey-Foster  method. 

99.  Measurement  of  Mutual  Inductances.1 — The  mutual 
inductance  of  two  coils  may  be  measured  in  terms  of  capacity 
and  resistance  by  means  of  a  method  due  to  Carey-Foster,  in 
which  the  quantity  of  electricity  induced  in  the  secondary  is 
balanced  against  a  known  charge  from  a  standard  condenser. 
The  connections  are  shown  in  Fig.  65,  where  P  and  S  are  the 

1  CAREY-FOSTER,  Phil.  Mag.,  vo.  23,  p.  121. 
CARHART  and  PATTERSON,  Electrical  Measurements,  p.  268. 
SMITH,  Electrical  Measurements,  p.  217, 


124  ELECTRICITY  AND  MAGNETISM 

primary  and  secondary  coils  of  the  mutual  inductance  to  be 
measured,  C  a  standard  condenser,  and  G  a  ballistic  galvanometer. 
The  primary  circuit  is  represented  by  the  path  BPARi,  while  the 
secondary  is  SR^DA  including  the  galvanometer  G.  It  will  be 
noted  that  the  galvanometer  is  also  included  in  circuit  DCRiA 
containing  a  standard  condenser.  When  the  primary  circuit  is 
closed  the  galvanometer  will  be  traversed  by  two  distinct  quanti- 
ties of  electricity:  (1)  The  quantity  induced  in  the  secondary  coil, 
and  (2)  the  charge  entering  the  condenser,  both  of  which  may 
easily  be  computed.  If  these  two  quantities  are  equal  and  pass 
through  the  galvanometer  in  opposite  directions,  no  deflection 
will  result,  which  is  the  balance  condition  sought. 

The  quantity  Qi  induced  in  the  secondary  coil  is  the  time 
integral  of  the  secondary  current,  during  the  interval  required  for 
the  primary  to  rise  from  zero  to  its  final  value  /. 
That  is, 


M  C1          MI 


where  R  is  the  effective  resistance  of  the  secondary  circuit.  The 
quantity  Q2  of  electricity  passing  through  the  galvanometer  to 
charge  the  condenser  is  given  by 

Q2  =  CV  =  CRJ  (14) 

where  V  =  RJ  is  the  fall  of  potential  across  R  i  which  is  charging 
the  condenser.  Equating, 


=  CRJ,  (15) 

M  =  CRiR.  (16) 

Since,  at  the  point  of  balance,  there  is  no  current  through  the 
galvanometer,  and  consequently  no  fall  of  potential  across  it, 
the  effective  resistance  R  of  the  secondary  circuit  includes  only 
Rz  and  S.  The  final  formula  then  becomes 

M  =  CR^Rz  +  S)  (17) 

If  C  is  expressed  in  farads,  and  the  resistances  in  ohms,  M  will  be 
given  in  henries. 

100.  Experiment  18.     Mutual  Inductance  by  the  Carey-Foster 
Method.  —  Connect  the  apparatus  as  shown  in  Fig.  65,  where  PS 


SELF  AND  MUTUAL  INDUCTANCE  125 

is  a  variable  mutual  inductance  whose  calibration  curve  is  to  be 
obtained,  C  a  subdivided  standard  condenser,  G  a  ballistic  gal- 
vanometer of  long  period,  and  B  a  storage  battery  of  20  volts. 
It  is  necessary  that  the  four  wires  indicated  at  A  should  actually 
meet  at  a  common  point,  so  a  connector  should  be  used.  Since 
a  large  voltage  is  connected  directly  across  RI  there  is  danger  of 
burning  it,  so  compute  the  minimum  resistance  which  may  be 
used,  allowing  a  maximum  power  consumption  of  4  watts  per 
coil.  To  make  sure  that  the  discharges  through  the  galva- 
nometer oppose  one  another  and  are  of  the  same  order  of 
magnitude,  try  them  first  separately;  that  is,  break  the  circuit  at 
C,  make  and  break  the  primary  circuit  and  note  the  direction  of 
the  galvanometer  deflection  at  the  make,  due  to  the  induced 
current  in  the  secondary.  Now  close  the  circuit  again  at  C, 
breaking  the  secondary  at  RZ)  and  note  the  deflection  at  make, 
which  is  now  due  to  the  charge  entering  the  condenser.  If  the 
deflection  is  in  the  same  direction  as  before,  reverse  the  connec- 
tions on  either  the  primary  or  secondary  coil.  Close  the  circuit 
at  Rz  and  obtain  abalance  varying  RI,  R^  and  C.  The  resistance 
of  the  secondary  coil  may  be  obtained  by  means  of  a  post-office 
box. 

Report. — 1.  Plot  mutual  inductance  in  millihenries,  as 
ordinates,  and  positions  of  coil  as  abscissas. 

2.  How  would  your  results  be  affected  if  you  had  interchanged 
primary  and  secondary  coils?  Explain. 


CHAPTER  X 
ELEMENTARY  TRANSIENT  PHENOMENA1 

101.  Time  Constant,  Circuit  Having  Resistance  and  Induc- 
tance.— When  an  E.M.F.  is  suddenly  impressed  on  a  circuit 
containing  resistance  only,  the  current  rises  instantly  to  a  definite 
value  determined  by  Ohm's  law.  If,  however,  the  circuit  con- 
tains inductance  as  well  as  resistance,  this  is  not  the  case,  for 
while  the  current  is  being  established,  it  produces  within  the  coil 
a  magnetic  flux  which  links  the  turns  of  the  coil.  Whenever  a 
change  occurs  in  the  flux  through  a  coil  there  is  induced  within  it 
an  E.M.F.  in  such  a  direction  as  to  oppose  the  change  which 
produced  it.  From  the  definition  of  self  inductance,  the  value 

of  this  counter  E.M.F.  is  L~r  where  L  is  the  coefficient  of  self 

inductance.     It  is  thus  seen  that  the  impressed  E.M.F.  is  opposed 
R  L  by  two  counter  E.M.F.'s;  one  due  to 

— \AAAAA/ 'TRHftHP — i  the  current  flowing  through  the  re- 
sistance and  the  other  due  to  the 
rising  current  in  the  coil.  Such  a 
circuit  is  represented  in  Fig.  66,  where 
the  resistance  R  and  the  inductance  L 
FIG.  66.— Circuit  containing  are  shown  separately,  although  they 

resistance  and  inductance.  .  . 

may   co-exist  in   the   coil.      Let   the 

value  of  the  current,  t  seconds  after  closing  the  key,  be  i.     Then 
by  Ohm's  law,  we  have 

B-K+L*  (1) 

This  is  a  differential  equation  and  can  not  be  solved  by  the 
ordinary  rules  of  algebra.     Dividing  through  by  R  and  letting 

E 

I  =   n  be  the  final  value  of  the  current,  we  have 

.   Ldi  ,~N 


1  BEDELL  and  CREHORE,  Alternating  Currents 
PIERCE,  Electric  Oscillations  and  Electric  Waves. 
STEINMETZ,  Transient  Phenomena. 

126 


ELEMENTARY  TRANSIENT  PHENOMENA 


127 


Separating  the  variables,  we  obtain 

di         R  ,. 

j .  =  T  dt 

I  —  i      L 


Integration  of  both  sides  gives 


R 


-log (/  - 1>  -  F  *  + 


(3) 


(4) 


where  C  is  an  arbitrary^constantf  whose  value  may  be  obtained  by 
substituting  corresponding  known  values  for  i  and  t.  Counting 
time  from  the  instant  the  key  is  closed,  when  t  =  o,  i  =  o,  and 
these  quantities  when  substituted  in  eq.  (4)  give  C  =  —log  /. 
Hence  eq.  (4)  becomes,  on  replacing  C  by  its  value  and  rearranging, 

.       (I  -i)  R  . 

log f —  =  -  T  t  (5) 


Taking  the  antilogarithm  of  both  sides, 

I  -i 


=  e 


-£< 


(6) 


where  e  is  the  base  of  the  Naperian  logarithms.     Solving  for  i,  we 
have 

t'-/(l  -e-l1  )  (7) 


Otitt       t3  t,  t 

FIG.  67. — Growth  of  current  in  a  circuit  containing  resistance  and  inductance. 

The  graph  of  this  equation  for  a  series  of  values  of  L  with  constant 
R  and  E  is  shown  in  Fig.  67.  It  is  seen  that  when  L  =  o  the 
last  term  of  eq.  (7)  vanishes  and  the  current  rises  immediately  to 
its  final  value ;  but  as  L  is  made  larger  a  longer  time  is  required  for 
it  to  reach  a  given  fraction  of  its  final  magnitude.  It  is  obvious 
that  inductively  wound  coils  might  be  classified  according  to  the 
time  required  for  the  current  to  reach  a  certain  specified  fraction 


128  ELECTRICITY  AND  MAGNETISM 

of  its  final  values  under  a  constant  impressed  E.M.F.  The  most 
suitable  fraction  to  choose  is  arrived  at  in  the  following  way. 

If,  ineq.  (7),  /  =  &>  there  results 

i  =  /(l  -    -)  =  .6327 

The  quantity  „  is  called  the  "Time  Constant"  for  the  coil  and  is 

defined  as  the  time  required  for  the  current  to  reach  .632  of  its 
final  value  under  the  action  of  a  constant  E.M.F.  The  values 
ti,  ^2,  £3,  etc.,  in  Fig.  67  represent  the  time  constants  for  the  various 
values  of  L. 

102.  Circuit    Having   Resistance    and    Capacitance. — A  case 

quite  similar  to  the  one  discussed 
above  is  that  in  which  an  E.M.F.  is 
suddenly  impressed  upon  a  circuit 
containing  resistance  and  capacitance 
in  series.  Such  an  arrangement  is 
shown  in  Fig.  68.  As  soon  as  the 
key  is  closed,  a  current  flows  through 
FIG.  68.— Circuit  containing  R  and  a  charge  begins  to  accumulate 

resistance    and    capacitance   in     ^  Q       ^^  ^^  ^    Q^Q    produces 

a  counter  E.M.F.,  which,   added  to 
that  due  to  the  current  through  R,  balances  the  impressed  E.M.F. 

The    potential  difference  across  the   condenser  is   ^  01 
Accordingly,  we  may  write 

E  =  Ri  +  ~  \m.  (8) 


It  is  more  convenient  to  solve  this  equation  in  terms  of  the 
instantaneous  charge  q  in  the  condenser  than  of  the  current 

through  the  resistance.     Remembering  that  i '  =  -37  we  have,  on 
substitution  in  eq.  (8), 

E  =  RM+C  (9) 

Multiplying  through  by  C  and  putting  CE  =  Q,  the  final  charge 
in  the  condenser,  eq.  (9)  becomes,  on  separating  the  variables, 

d(l       .    di  /10\ 

Q=-q-RC 


ELEMENT AR  Y  TRANSIENT  PHENOMENA 


129 


Integrating  both  sides  of  eq.  (10),  we  have 
-log  (Q  -  «)  =  ~  +  K 


(11) 


As  before,  K  is  an  arbitrary  constant  of  integration  which  may  be 
evaluated  by  subtistuting  known  values  of  q  and  t  in  eq.  (11). 
Counting  time  from  the  instant  of  closing  the  key,  we  have,  when 
t  =  o,  q  =  o.  Substituting  in  ejq.  (11) 

K  =  -logQ 
Replacing  K  by  its  value,  and  rearranging  terms,  eq.  (11)  becomes 


! 

°g 


(12) 


(13) 


(14) 


This  equation  is  analagous  to  eq.  (7)  of  the  previous  article  and 
its  graph  is  shown  in  Fig.  69,  for  several  values  of  R  with  constant 


Q  RC 

Taking  the  antilogarithm  of  both  sides,  we  have 

V9=^ 

Solving  for  q,  there  results 

_  j_ 

.  -  i*c) 


FIG.  69. — Growth  of  charge  for  a  circuit  containing  resistance  and  capacitance. 

E  and  C.  If  R  =  o,  the  condenser  becomes  charged  instantly 
to  its  final  value  Q,  but  when  a  series  resistance  is  included,  a 
definite  time  is  required  for  the  condenser  to  become  charged.  Such 
circuits  may  be  classified  according  to  the  time  required  for  the 
charge  to  reach  a  specified  fraction  of  its  final  value.  As  before 
this  fraction  is  arrived  at  by  putting  t  =  RC. 


130  ELECTRICITY  AND  MAGNETISM 

Eq.  (14)  then  becomes 

q  =  QI  -.1    =  .632  Q. 


The  quantity  RC  is  called  the  time  constant  for  a  circuit 
containing  resistance  and  capacitance,  and  is  defined  as  the  time 
required  for  the  charge  to  reach  .632  of  its  final  value.  These 
times  are  shown  for  the  successive  values  of  R  by  ti,  ti,  tS)  etc.,  in 
the  figure.  The  time  constant  is  an  important  concept  in  the 
study  of  reactive  circuits  and  will  be  referred  to  frequently  in 
this  text  in  the  discussions  to  follow. 

103.  Circuit  Containing  Resistance,  Inductance  and  Capaci- 
tance. Discharge  of  a  Condenser. — To  describe  some  of  the  phe- 
nomena peculiar  to  a  circuit  containing  resistance,  inductance 

and  capacitance,  it  will  be  supposed 
^  A  A  /^OOOOTN  1 1 c  ^nat  the  parts  are  connected  in  series 


condenser   has  been  charged  by  ap- 
K  propriate    means.     Suppose    further 

that    the  key    has   been   closed  and 

FIG.  70.— Circuit  containing     that  it  is  discharging;  also  that  the 

instantaneous  current  is  i  and  the 
charge  in  the  condenser  is  q.  Since 
no  external  E.M.F.  is  acting,  the  sum  of  the  differences  of  poten- 
tial across  the  three  elements  of  the  circuit  must  be  zero  at  all 
times.  Accordingly, 

U  =  0.  (14) 

Differentiating  and  dividing  through  by  L  we  have 

d^    ,R<tt_       _L  -  =  o 
dt2  +  L  dt  +  LC  * 

Since  i  =  -jb  eq.  (14)  may  also  be  written 


eqs.  (15)  and  (16)  are  sufficient  to  completely  describe  a  circuit 
of  this  character.  Since  they  are  identical,  only  one  of  them, 
e.g.,  (15),  will  be  discussed. 

This  is  a  linear  differential  equation  of  the  second  order  with 
constant  coefficients  and  may  be  solved  in  the  following  manner : 
Let 

i  =  kemt  (17) 


ELEMENTARY  TRANSIENT  PHENOMENA  131 

where  k  is  an  arbitrary  constant  depending  upon  the  boundary 
conditions  and  ra,  another  constant,  depending  upon  the  coeffici- 
ents of  the  original  differential  equation.  Differentiating  eq.  (17) 
twice  and  substituting  in  eq.  (15),  there  results 

«2+|m+z^  =  0.  (18) 

This  equation  gives  t]ie  valued  that  must  be  assigned  to  m  in 
order  that  eq.  (17)  may  be  the  solution  of  eq.  (15).  Solving, 

-RC  +  VR2C2  -  4LC 

""  (19) 


It  is  thus  seen  that  there  are  two  values  of  m  which  will  make  eq. 
(17)  a  solution  of  eq.  (15).  These  give  what  are  known  as  "  par- 
ticular solutions"  and  the  "  complete  solution"  is  obtained  by 
adding  them  together.  Accordingly, 


rRC  -  \/R*C*  -  4LCi  rRC 


i  =  ktf  2LC  +  k2e  c  (20) 

The  solution  for  q  is  identical  except  that  different  arbitrary 
constants  will  appear.  Call  them  k3  and  &4.  It  is  to  be  noted 
that  the  coefficient  of  t  in  the  exponential  term  contains  a  radical, 
the  quantity  under  which  may  be  positive,  zero,  or  negative 
according  to  the  relative  values  of  R,  L,  and  C.  The  theory  of 
differential  equations  shows  that  the  character  of  the  solutions 
under  these  circumstances  is  quite  different,  and  that  we  have 
three  distinct  cases  to  consider. 

Case  I.  R2C2>4LC.  Non-oscillatory  Discharge.  —  For  simplic- 
ity, let 

_  2LC  _  2LC 

Tl  ~  RC-  V~R*C*  -  4LC  a  =  RC  +  VR2C*  -  4LC     (21' 

The  solutions  of  eqs.  (15)  and  (16)  may  then  be  written 

_1  _1 

i  =  kie   TI  +  kze   ^  (22) 

_1  _! 

q  =  kse   'I  +  k*e   ^  (23) 

TI  and  T2  are  thus  seen  to  be  time  constants  and  it  is  to  be  noted 
that  when  both  inductance  and  capacity  are  present,  the  circuit 
possesses  two  time  constants  instead  of  one  as  in  the  cases 
previously  considered.  The  arbitrary  constants  ki,  fc2,  &3,  &4 
may  be  determined  in  the  following  way.  If  time  is  reckoned 
from  the  instant  the  key  is  closed,  then  when 

t  =  0,  i  =  0,  q  =  Q  (24) 


132  ELECTRICITY  AND  MAGNETISM 

Substituting  these  values  in  eqs.  (22)  and  (23)  there  results ' 

0  =  ki  +  kz     Q  =  ka  +  &4  (25) 

Differentiating  eq.  (23) 

f  -  *  =  -V'';  -  V-'.  (26) 

tt£  TI  T2 

Comparing  coefficients  in  eqs.  (22)  and  (26)  we  have 

fa  =  -^  and  k2  =  -^  (27) 

Tl  T2 

Substituting  the  values  of  7c3  and  fc4  from  eqs.  (27)  in  (25)  and 
eliminating,  the  following  values  are  obtained: 


T2    —  Tl  TI    —   T2 

fcs  =  T^V2  ^-^V,       .  (») 

Substituting  these  values  in  eqs.  (22)  and  (23)  we  have 


T2 


e   *•  -  e   *.  |  (29) 

JL_  rie~^  _  r^%,l  (30) 

-T2|_  J 

It  is  thus  seen  that  the  solutions  are  made  up  of  two  exponential 

curves  whose  difference  is  to  be  taken. 
In  the  case  of  the  current,  these  curves 
have  the  same  initial  ordinates  but 
approach  the  time  axis  at  different 
rates  because  of  the  different  time 
constants.  The  solution  is  shown 
graphically  in  Fig.  71,  where  the 

FIG.  71.— Aperiodic  discharge     dotted    curves   are   the   separate  ex- 
ponentials and  the  full  line  represents 

their  difference.     The  current  starts  at  zero,  rises  to  a  maximum 

and  then  slowly  dies  away. 

Case    II.  R2C2  =  4   LC.     Critically   Damped   Discharge. — In 

7"> 

this  case  the  roots  of  eq.  (18)  are  identical  having  the  value  —  2> 

and  the  two  terms  of  eq.  (22)  are  the  same.  This  equation 
cannot  be  the  complete  solution  for  this  case  since  it  contains 
but  one  arbitrary  constant,  whereas  the  complete  solution 
must  have  two,  since  the  original  differential  equation  is  of  the 
second  order. 


ELEMENTARY  TRANSIENT  PHENOMENA  133 

The  theory  of  differential  equations1  shows  that  for  this  case 
the  solutions  of  eqs.  (15)  and  (16)  are 


_ 

2L  2L 

SE'  (32) 


Imposing  the  same  boundary  conditions  as  before,  namely,  when 
t  =  0,  i  =  Ot  and  q  =~Q,  we   Have 


ki  =  0  and  ks  =  Q 
Differentiating  eq.  (32) 

.  _  dq        _  k$R    -  2Ll     |    T  I    -  2Z/        jR.    "<§;']    (QQ\ 

L  I 

Applying  the  first  boundary  condition  to  eq.  (34)  gives 

I.     ~D  f\  p 

n/3/V  ^J-tt 

4  =  or    ==  ^F 

.4-L/  -  /> 

Comparison  of  coefficients  in  eqs.  (31)  and  (33)  gives 
R  ,  QR*  _    _E 

2L    4  =         4L2~       L 

The  complete  solutions  accordingly  are 

i=  -^te  ~^1  (34) 

Kt 

2L  (35) 

These  equations  consist  of  the  product  of  a  straight  line  and  an 
exponential  curve,  and  are  similar  to  the  corresponding  ones  for 
Case  I.  If  numerical  values  are  substituted,  it  is  found  that  they 
rise  to  higher  values  and  that  they  are  more  compressed  along 
the  time  axis.  In  fact,  the  theory  shows  that  for  this  critical 
case  the  discharge  takes  place  in  the  shortest  time  possible. 
Case  III.  R2C2<4LC.  Oscillatory  Discharge.— This  is  the 
most  interesting  and  important  of  the  three  cases.  The  quantity 
under  the  radical  sign  of  eq.  (19)  then  becomes  imaginary  and  the 
two  roots  of  eq.  (18)  are  complex  quantities.  Call  them 

mi  =  a  +  j/3  and  m2  =  a  —  jfi 
where 

Ki     n         'V'  4.L/U         Ki  O  i     .  /      i 

— ,  and  j  =    v  —1. 


MURRAY.  Differential  Equations,  p.  65. 


134  ELECTRICITY  AND  MAGNETISM 

Equation  20  may  then  be  written 


Let 

A  ~~ 


eak        +  k&-    \ 

eat[ki(co$  pt  +  j  sin  fit)  +  k2  (cos  pt  -  j  sin  pi)] 

e**[(ki  +  fc2)  cos  pt  +  (ki  -  fc2)  j  sin  #] 


then  fci  +  fa  =  A 


whence 

i  =  eat[A  cos  #  +  B  sin  /ft] 

By  means  of  a  well  known  formula  of  trigonometry  this  may  be 
written 

i  =  keai  sin  (pt  +  </>)  (36) 

where 

A 
k  =  V^2  +  B2  and  0  =  tan-1  # 

In  a  similar  manner  the  solution  of  eq.  (16)  for  this  case  may  be 
shown  to  be 

q  =  k'eat  sin  (pt  +  <£')  (37) 

The  four  arbitrary  constants  k,  k',  0,  <$>'  are  real  quantities  and  may 
be  determined  by  imposing  the  same  boundary  conditions  as 
used  above.  Substituting  in  eqs.  (36)  and  (37)  the  values  i  =  0, 
q  =  Q  for  t  =  0  respectively  they  become 

0  =  k  sin  $          whence  0=0  (3.8) 

Q  =  k'  sin  0'  k'  =  -J^-f 

sin  0 

Differentiating  eq.  (37)  with  respect  to  t 

i  =        =  k'e*  [a  sin  (pt  +  00  +  P  cos  (pt  +  0')] 


(39) 

J 

Using  again  the  condition  i  =  0  for  t  =  0  we  have 

R 

tan  <£'  =  -  - •  = 
Q  Q 


_  _    ^ 

sin  0'          .  V4LC  -  #2C2      \/4LC  - 

sm  tan-1        ~— 


ELEMENTARY  TRANSIENT  PHENOMENA  135 

Comparing  the  coefficients  of  the  sine  terms  in  eqs.  (39)  and  (36) 
we  have 

,  2Q 

-  vyf+f  =  vlzF^W* 

The  complete  solutions  may  nqw  be  written 


.   V4LC  -4  ,.n. 

-~ 


- 

82Lc 

.xALC-iwl    ._, 
tan       -fie  ----  J   (41) 

The  current  and  charge  are  sine  functions  of  the  time  and  are 
therefore  oscillatory  in  character.  The  initial  amplitude  of  the 
oscillations  is  proportional  to  the  charge  given  to  the  condenser 


FIG.  72. — Damped  sine  wave. 

and  depends  also  upon  the  constants  R,  L,  and  C  of  the  circuit. 
Furthermore,  the  amplitude  is  multiplied  by  an  exponential 
factor  which  decreases  with  the  time  and  the  oscillations  conse- 
quently die  out.  An  oscillation  of  this  character  is  spoken  of  as  a 
" damped"  sine  wave.  The  graph  for  the  current  wave  is  shown 
in  Fig.  72.  That  for  the  charge  is  similar  to  it  except  that  its 
phase  is  ahead  of  the  current  by  the  angle  whose  tangent  is  given 
by  the  last  term  in  eq.  (41).  If  R  =  0,  this  angle  is  90°. 

The  period  T  of  the  oscillation  is  obtained  from  eq.  (40)  by  the 
relation 

A/4LC  -  R*C*  _  27r 
2LC  ~  T 


136  ELECTRICITY  AND  MAGNETISM 

whence 


If  R2C2  may  be  neglected  in  comparison  to  4LC,  this  reduces 
to  the  simple  expression 

T  =  2ir\/LC  (42) 

104.  Logarithmic  Decrement. — The  physical  interpretation  of 
the  phenomenon  just  described  in  mathematical  terms  is  as 
follows :  When  the  condenser  is  given  a  charge,  a  definite  amount 
of  energy,  %C  F2,  is  stored  up  in  it.  As  it  discharges  and  current 
flows  through  the  circuit,  this  energy  is  in  part  dissipated  by  the 
resistance  R  and  in  part  stored  up  in  the  electromagnetic  field  of 
the  inductance  L.  At  the  instant  the  potential  difference  across 
the  condenser  is  zero  the  energy  which  has  not  been  dissipated  as 
heat  is  in  the  coil  has  the  value  J^L/2.  This  energy,  minus 
that  dissipated  during  the  next  quarter  swing  is  returned  to  the 
condenser  charging  it  in  the  opposite  direction  and  So  on.  If 
the  circuit  were  entirely  free  from  resistance,  the  oscillations  would 
simply  represent  interchanges  of  energy  between  the  condenser 
and  the  coil  at  a  frequency  twice  that  of  the  circuit  and  would 
continue  indefinitely  in  much  the  same  manner  as  a  pendulum 
suspended  by  frictionless  bearings  in  a  vacuum.  It  is  obvious 
that  the  greater  the  rate  of  energy  dissipation,  the  smaller  the 
number  of  oscillations.  The  quantitative  method  of  treating 
the  damping  effect  is  as  follows: 
Write  eq.  (40)  in  the  simplified  form 

i  =  Ie-at  sin  wt  (43) 

where 

T  =          ___2_£  R  V±LC  -  R2C* 

=  V4LC  -  #2C2 "       2L    '  2LC 

Let  /i,  /2,  Iz In  be  the  successive  current  maxima  as  indi- 
cated in  Fig.  72,  let.^i,  £2,  t3 tn  be  the  times  at  which  they 

occur,  and  let  T  be  the  period  of  oscillation.     Since 

sin  cotfi  =  sin  ut<>  =  .  .        .  =  1 


J,  = 


In  = 


ELEMENTARY  TRANSIENT  PHENOMENA  137 

The  ratio  of  the  first  amplitude  to  any  succeeding  one  is 

*l=6«<»-l>r  (44) 

J-n 

In  particular,  let   n   =  2.     Then 


It  is  easily  seen  that  4he  ratio*bf  any  amplitude  to  the  next  one 
succeeding  it  is  constant  and  is  equal  to  the  value  just  given. 
Taking  the  logarithm  of  both  sides 


log,  £  =  O.T  =  irR  Vf  =  8 

1  2  JW 


(45) 


The  quantity  5  is  called  the  "  Logarithmic  Decrement  "  and 
is  defined  as  the  Naperian  logarithm  of  the  ratio  of  any  amplitude 
to  the  next  one  succeeding  it  in  the  same  direction,  and  is  given 
in  terms  of  the  constants  of  the  circuit  by  eq.  (45).  One 
of  the  many  applications  that  may  be  made  of  this  quantity  is 
the  determination  of  the  number  of  oscillations  that  the  circuit 
will  execute  before  the  amplitude  is  reduced  to  an  assigned  frac- 
tion of  its  initial  value.  For  example,  between  I\  and  /n+i,  there 
are  n  oscillations. 
Substituting  in  eq.  (44) 


n  +  l 

or 


whence 
where 


n  =  \ 


Tn+1  is  the  assigned  fraction. 

i  n 


105.  Harmonic  E.M.F.  Acting  on  a  Circuit  Containing  Resis- 
tance, Inductance  and  Capacitance. — The  equation  of  E.M.F.'s 
for  this  case  is 

L  ™+  Ri  +  ^  fwft  =  E  sin  «*.  (46) 

at  LJ 

The  theory  shows  that  when  the  second  member  of  a  differential 
equation  is  different  from  zero,  the  complete  solution  is  made  up 
of  two  parts :  (a)  the  solution  of  the  original  differential  equation 
when  the  second  member  is  put  equal  to  zero,  and  (b)  the  par- 


138  ELECTRICITY  AND  MAGNETISM 

ticular  integral.  Part  (a)  has  already  been  discussed  and  it  was 
found  to  represent  a  transient  phenomenon  which  quickly 
dies  out.  Part  (6)  corresponds  to  a  "forced"  oscillation,  and 
represents  a  steady  state.  It  is  the  part  in  which  we  are  inter- 
ested in  problems  of  continuous  alternating  currents. 

The  student  familiar  with  differential  equations  will  remember 
that  equations  of  the  form  of  (46)  are  best  treated  by  means  of 
the  'differential  operator  "D."  To  apply  this,  first  differentiate 
eq.  (46)  with  respect  to  t  to  remove  the  sign  of  integration 

dH  .    R  di   .     1     .        Eco 

W  +  Ldt+LC1-    ^cos^  (47) 

Introducing  the  operator  D 


The  particular  integral  which  we  are  seeking  is  then 

1  7T 

i  =  -      —5  --  cos  at  (49) 


The  meaning  of  the  " inverse"  operator,  as  the  quantity 
immediately  following  the  equality  sign  is  called,  is  this:  Find 
a  function  of  i  such  that  when  operated  on  by  the  coefficient  of 
i  in  eq.  (48)  it  gives  the  right-hand  member  of  that  equation. 
There  is  a  well  known  short  method1  for  treating  the  case  of  sines 
or  cosines  such  as  eq.  (49).  It  consists  simply  in  expressing  the 
function  of  D  as  a  function  of  D2  and  replacing  D2  by  minus  the 
square  of  the  coefficient  of  the  independent  variable.  Accord- 
ingly 

1  Ea  Ea 

I   =  ^ :       -=r-  COS  at  = COS  0)1  = 


[RD  -  (±  - 


cos  at 


EV[RD  -g  - 


MURRAY,  Differential  Equations,  p.  77. 


ELEMENTARY  TRANSIENT  PHENOMENA  139 

sin  co* 


^«l7t  —  Leo2  j  cos  eoi 
/I  \ 

7~>O       Oil  T        O  1 

/C2co2+  (77  —  Lco2j 


cos  co£ 


combining  into  a  single  sine  function 
E 


i  =  -  ^T^" sm 

C^o 


+  (Lc°  ~  ~L 


co*  —  tan"1 


(50) 


It  is  thus  seen  that  the  current  is  a  sine  function  and  has  the 
same  frequency  as  the  impressed  E.M.F.  In  general,  it  is  not  in 
phase  with  the  E.M.F.  but  lags  behind  or  leads  according  as 

Leo  is  greater  or  less  than  ~ —     If  Leo  =  7^—,  i.e.,  eo  =  -  -j=->  the 

Ceo  Ceo  VLC 

current  is  in  phase  with  the  E.M.F.  and  in  this  case  eq.  (50) 
becomes 

.       E   . 
i  =  -=5  sin  co* 
ri 

which  is  identical  with  the  current  equation  given  directly  by 
Ohm's  law  for  the  case  when  no  inductance  or  capacitance  is 
present.  The  maximum  value  of  the  current  is  obtained  by 
putting  the  sine  function  equal  to  unity:  i.e., 

i=        E 


By  analogy  with  Ohm's  law  the  denominator  is  called  the  "  Impe- 
dance" of  the  circuit  and  the  quantities  Leo  and  ^-  are  called 

uco 

the  inductive  and  capacitive  " Reactances"  respectively.  Reac- 
tance produces  not  only  a  phase  angle  between  the  current  and 
E.M.F.  but  also  reduces  the  magnitude  of  the  current. 

106.  Alternative  Method. — For  those  unfamiliar  with  differ- 
ential equations,  an  indirect  method  of  obtaining  the  solution  of 
eq.  (46)  may  be  employed.  Since  an  alternating  E.M.F.  is  applied 
to  the  circuit,  it  is  reasonable  to  suppose  that  the  current  will  also 
be  alternating,  that  it  will  have  the  same  frequency  as  the  E.M.F. 


140  ELECTRICITY  AND  MAGNETISM 

and  that  it  may  not  be  in  phase  with  the  E.M.F.  These  assump- 
tions are  combined  in  the  following  expression 

i  =  I  sin  (co£  +  <f>)  (51) 

where  /  and  <£  are  arbitrary  constants  which  are  to  be  determined 
by  substituting  eq.  (51)  in  (46)  and  finding  the  values  which  must 
be  assigned  to  them  in  order  that  eq.  (46)  may  be  satisfied. 

r  =  /co  cos  (ut  +  <£)  and  fidt  =  --  cos  (coZ  +  <t>) 
at  co 

Substituting  these  .values,  eq.  (46)  becomes 

L/co  cos  (coZ  +  <£)  +  RI  sin  (coZ  +  0)  —77-  cos  (co£  +  <£) 

Ceo 

Since  this  equation  holds  for  all  values  of  t,  we  may  write, 
when 

at  4-  <f>  =  0,  L/co  —  7r  =  —E  sin  <£ 
Ceo 

when  co£  +  <£  =  _>  RI  =  E  cos  <£ 

Squaring  and  adding  the  above  expressions, 

* 

E 


./= 


Dividing  one  by  the  other 


—  tan  0  =  -   —  f:  -  or  <t>  =  —tan" 


-p  \Ji  \p   -  p 

Substituting  these  values  in  eq.  (51)  we  have 
E 


==  sin 

2 


-  tan-1 


R 


which  is  eq.  (50)  above. 

107.  Vector  Diagrams. — In  the  discussion  thus  far,  we  have 
spoken  of  alternating  E.M.F.'s  and  alternating  currents  and  have 
used  in  each  case  the  trigonometric  expressions  in  discussing 
them.  For  example  the  equations 

e  =  E  sin  co£ 

i=I  sin    co£  —  0 


ELEMENTARY  TRANSIENT  PHENOMENA 


141 


have  been  used  to  represent  respectively  an  alternating  E.M.F. 


having  a  maximum  value  J£  and  a  frequency/  =  ^-'  and  an  alter- 

nating current  of  the  same  frequency  with  a  maximum  value  1 
lagging  behind  the  E.M.F.  by  a  phase  angle  <£. 

These  may  be  regarded  as  being  given  by  the  projections  on  the 
Y  axis  of  the  vectors^0#  and^O/  respectively  of  Fig.  73  which 
rotate  with  constant  angular  velocity  in  counter  clockwise  direc- 
tion, the  latter  lagging  behind  the  former  by  the  angle  <£.  The 
vectors  OE  and  01  represent  the  maximum  values  of  the  E.M.F. 


FIG.  73. — Sine  waves  represented  by  rotating  vectors. 

and  current.  As  a  special  case,  consider  that  of  an  alternating 
E.M.F.  acting  on  a  circuit  having  resistance  and  inductance.  The 
current  is  given  by  eq.  (50)  with  C  =  «> ,  the  condition  for  zero 
capacitive  reactance.  Thus 

•  J7F  7- 

flj  /  \j(jj\ 

sinl  ut  —  tan"1  -=-  I  (52) 

\  zt  / 


For  the  maximum  value  of  the  current,  we  have 

Tjl 

-  or  E  =  \/#2/2  +  L2o>2/2 


L2C02 

From  the  form  of  the  latter  expression  it  is  evident  that  E  has 
such  a  value  that  it  may  be  given  as  the  diagonal  of  a  rectangle 
whose  sides  are  RI  and  Lo?7,  as  shown  in  Fig.  74.  The  current  / 
is  represented  as  a  vector  in  phase  with  RI,  since,  from  eq.  (52), 
the  current  lags  behind  the  E.M.F.  by  an  angle  whose  tangent  is 

-p  •    This  is  the  angle  <f>  shown  in  the  figure.     If  this  figure  is 

rotated  about  the  origin  0  with  an  angular  velocity  w  in  the  posi- 
tive direction,  the  projections  of  the  vectors,  E,  RI,  and  Lwl  upon 
the  Y  axis  give  the  instantaneous  values  of  the  impressed  E.M.F. 


142 


ELECTRICITY  AND  MAGNETISM 


and    the    E.M.F.    across    the    resistance    and   the   inductance 
respectively. 

In  a  similar  manner,  a  vector  diagram  may  be  constructed  for 
a  circuit  containing  resistance,  inductance  and  capacitance  in 
series.  For  this  case  the  maximum  current  and  phase  angle  are 
given  respectively  by 

7-  E 


=  tan-1 


Lul 


O  RI 

FIG.  74. — Vector  diagram  for  resistance 
and  inductance. 


Lul 


I 

cu 


RI 


FIG.  75. — Vector  diagram  for  resis- 
tance, inductance  and  capacitance. 


The  vectors  are  similar  to  those  of  the  previous  case  except 
for  the  additional  vector,  ~-,  which  is  shown  drawn  downward  in 
Fig.  75,  since  in  the  equation  it  appears  as  a  quantity  subtracted 

from  Leo.     In  the  figure,  Leo/  is  shown  greater  than  -~-  and  the 

Ceo 

current  lags  behind  the  E.M.F.     If  Leo  =  ~->  the  component  of  E 

Ceo 

perpendicular  to  /  is  zero  and  the  current  is  in  phase  with  the 
E.M.F.     On  the  other  hand  when  ^-  is  greater  than  Leo,  $  is 

negative,  and  the  current  leads  the  E.M.F. 

108.  Electrical  Resonance. — In  discussing  the  discharge  of  a 
condenser  through  a  circuit  containing  resistance  and  inductance, 
it  was  shown  that  when  the  resistance  is  less  than  a  certain  critical 


ELEMENTARY  TRANSIENT  PHENOMENA 


143 


value,  oscillations  occur.  If  such  a  circuit  is  acted  upon  by  an 
alternating  E.M.F.  whose  frequency  is  the  same  as  the  natural 
frequency  of  the  circuit,  alternating  currents  of  large  amplitude 
are  set  up  in  the  inductance  and  condenser.  This  phenomenon  is 
spoken  of  as  electrical  resonance  and  is  analogous  to  the  motion  of 
a  mechanical  system  possessing  inertia  and  elasticity,  when  acted 
upon  by  an  alternating  mec^nical  force  having  a  frequency 
corresponding  to  its  own  free  period.  Two  distinct  cases  occur 
depending  upon  whether  the  inductance  and  capacitance  are  in 
series  with  the  E.M.F.  or  are  connected  across  it  in  parallel. 
These  are  distinguished  as  "Series  Resonance"  and  "Parallel 
Resonance"  respectively. 

109.  Series  Resonance. — This  case  has  been  discussed  above  in 
some  detail.  The  instantaneous  value  of  the  current  must 
satisfy  eq.  (46)  the  solution  of  which  is  eq.  (50).  The  amplitude 
of  the  current  is  given  by 

7=  E 


and  it  has  already  been  pointed  out  this  is  a  maximum  when 

Leo  =  rf-'  which  is  the  condi- 
Co> 

tion  for  resonance.  The  cur- 
rent is  then  given  by  E 
divided  by  R  as  required  by 
Ohm's  law.  The  resonance 
condition  depends  upon  the 
relative  values  of  L,  C  and  co, 


E 


FIG.  76. — Series  resonance. 


FIG.  77. — Effect  of  resistance  on  sharp- 
ness of  resonance. 


and  may  be  brought  about  by  a  suitable  change  of  either  one  of 
them,  the  other  two  being  held  constant.  Bringing  a  circuit 
into  resonance  is  generally  spoken  of  as  " tuning"  it. 

The  dependence  of  the  current  upon  the  constants  of  the 


144  ELECTRICITY  AND  MAGNETISM 

circuit  may  be  illustrated  by  the  curves  shown  in  Fig.  77,  where 
the  current  amplitude  is  shown  as  a  function  of  frequency  for  a 
short  range  each  side  of  resonance.  The  inductance  and  capaci- 
tance are  held  constant  and  three  different  resistances  are 
indicated.  A  represents  the  current  at  resonance  for  a  small 
resistance  and  C  that  for  a  large.  It  is  to  be  noted  that  the  effect 
of  a  change  in  resistance  is  much  more  marked  at  resonance  than 
at  a  frequency  somewhat  removed.  This  is  because  at  reson^ 
ance,  resistance  alone  determines  the  current,  while  at  low  fre- 
quencies, the  capacity  reactance  -~r  is  an  important  term,  but  at 

Ceo 

high  frequencies,  the  inductive  reactance  Leo  becomes  effective  in 
reducing  the  current.  It  is  to  be  noted  also,  that  for  low  fre- 
quencies the  current  leads  the  E.M.F.,  is  in  phase  with  it  at 
resonance  and  lags  behind  at  high  frequencies.  When  the  resis- 
tance is  small,  the  rate  of  change  of  the  phase  angle  in  passing 
through  resonance  is  rapid. 

110.  Parallel  Resonance.1 — When  the  E.M.F.  is  introduced  in 
the  circuit  in  such  a  way  that  the  inductance  and  condenser  are  in 

parallel,  the  phenomena  are  strik- 
ingly different  from  those  of  the 
series  arrangement  just  described. 
The  connections  for  this  case  are 
shown  in  Fig.  78.  Assuming  that 
the  condenser  is  free  from  energy 
absorption,  the  current  through  it 
leads  the  E.M.F.  by  ninety  degrees, 
while  that  through  the  inductance 
lags  behind  by  an  angle  depending 
upon  R,  L,  and  co.  The  current  in 
FIG.  78.— Parallel  resonance.  the  main  circuit  is  the  vector  sum 

of  these  two  and  in  determining  it 

the  relative  phases  of  the  components  must  be  taken  into 
account.  Denoting  the  currents  through  the  inductance  and 
condenser  by  IL  and  Ic  respectively,  their  amplitudes  are  obtained 
from  eq.  (50)  as  follows 

IL  =       /j= ==          Ic  =  ECu  (53) 

V  R2  +  L2co2 
Letting  the  vector  OE  of  Fig.  79  represent  the  impressed  E.M.F., 

1  Circ.  74.  U.  S.  Bureau  of  Standards,  p.  39. 


ELEMENTARY  TRANSIENT  PHENOMENA 


145 


the  above  currents  are  given  by  OIC  and 
OIL  respectively,  and  the  resultant  cur- 
rent 07,  is  the  diagonal  of  the  paral- 
lelogram formed  by  them  as  sides.  The 
amplitude  of  the  resultant  current,  by 
the  law  of  cosines,  is:  , 


72  = 


(54) 


The  value  of  cos  \f/  may  be  obtained  by 
remembering  that  the  E.M.F.  across 
the  coil  is  made  up  of  two  parts: 
That  across  the  resistance,  RIL,  and 
that  across  the  inductance  LuIL.  The 
former  is  in  phase  with  IL  and  the  latter, 
ninety  degrees  ahead  of  it.  Accordingly 


cos  ^  = 


Leo/, 
E 


(55) 


Substituting   eqs.    (53)    and    (55)  in  (54) 

and  combining  we  have 


=  #2 


C2co2 


1 


FIG.    79. — Vector   diagram 
for  parallel  resonance. 


2CcoLco 

R2  +  L2co2       R*  +  L2c 


(56) 


Multiplying  numerator  and  denominator  of  the  second  term  by 
#2  +  L2o>2,  eq.  (56)  may  be  written 


Leo 


+    -- 

r  (R*  +ZAo2) 


(57) 


Equation  (57)  is  the  general  expression  for  the  current  drawn 
from  the  supply.  The  condition  that  this  current  should  be  in 
phase  with  the  driving  E.M.F.  is 


Substituting  the  values  from  eqs.  (53)  and  (56)  there  results 

Leo 


Cw  = 


R2  +  L2  eo' 


(58) 


The  value  of  eo  obtained  from  this  equation  is  not  exactly  that 
corresponding  to  the  natural  period  of  the  circuit  but  approxi- 
mates it  closely.     If  R  is  zero,  it  corresponds  exactly.     Introduc- 
10 


146 


ELECTRICITY  AND  MAGNETISM 


ing  this  condition  in  eq.  (57)  it  is  seen  that  the  current,  when  in 
phase  with  the  E.M.F.,  is 

ER 


I  = 


(59) 


It  is  important  to  note  that  for  small  values  of  R,  I  is  nearly 
proportional  to  R  and  that  if  R  were  zero,  /  would  also  be  zero. 
We  thus  have  the  extraordinary  situation  in  which  the  larger 
the  resistance  the  larger  the  current.  Figure  80  shows  the  varia- 
tion of  current  with  fre- 
quency in  the  neighbor- 
hood of  resonance.  It  is 
interesting  to  note  that  in 
the  case  of  series  resonance 
the  individual  voltages 
across  the  condenser  and 
coil  exceed  the  total  volt- 
age across  the  two  com- 
bined, while  in  parallel 
resonance,  the  current  in 
each  exceeds  the  two  com- 
bined. The  series  arrange- 
ment gives  a  low  imped- 
ance at  resonance,  while 


FIG.  80. — Dependence  of  current  on  fre- 
quency for  parallel  resonance. 


the  parallel  connection  gives  a  high  impedance  at  this  point. 
For  this  reason,  the  latter  is  frequently  inserted  in  a  circuit 
when  it  is  desired  to  suppress  a  particular  frequency  in  a  complex 
wave. 

111.  Measurement  of  Inductance  and  Capacitance  by  Reso- 
nance.— The  phenomenon  of  electrical  resonance  furnishes  a 
convenient  method  for  the  determination  of  inductance  and 
capacitance,  particularly  when  they  are  small.  If  two  circuits, 
adjusted  to  have  the  same  natural  periods  are  placed  in  inductive 
relation,  and  one  of  them  is  caused  to  oscillate,  the  other  will 
oscillate  also  by  resonance.  It  was  shown  on  page  136  that  the 
period  of  an  oscillating  circuit  is  given  by  the  expression 

t  =  2ir\/LC. 

Consequently,  the  condition  for  resonance  is  that  the  LC  prod- 
ucts for  the  two  circuits  must  be  the  same  or 


=  L2C 


22 


(60) 


ELEMENTARY  TRANSIENT  PHENOMENA 


147 


where  the  subscripts  refer  to  the  circuits  1  and  2  respectively. 
If  three  of  these  quantities  or  one  LC  product  and  either  L  or  C 
are  known,  the  fourth  may  be  computed.  In  carrying  out  the 
measurement  it  is  more  satisfactory  to  use  a  third  circuit  as  a 
source  of  oscillations,  and  then  adjust  both  the  standard  and 
unknown  circuits  to  resonate  to  it.  The  inductance  and  capaci- 
tance of  the  third  circuit  should  be  adjustable,  but  need  not  be 
known.  The  three  circuits  are  shown  in  Fig.  81.  The  source 
circuit  is  energized  by  means  of  the  battery  A  and  an  ordinary 
buzzer  B  which  serves  as  an  interrupter.  When  the  armature 


-HIKJC^- 


C3 


0000000 


L-^n-O(>- 


FIG.  81. — Circuits  arranged  for  electrical  resonance. 


of  the  buzzer  closes  the  circuit,  the  battery  current  flows  through 
the  coil  L3  and  stores  up  energy  in  the  electromagnetic  field  link- 
ing its  windings.  When  the  armature  of  the  buzzer  breaks  the 
battery  circuit,  this  energy  is  transferred  back  and  forth  between 
C3  and  L3  until  it  has  been  dissipated.  A  group  of  damped  oscil- 
lations is  thus  established  in  this  circuit  for  each  vibration  of  the 
buzzer  armature.  Similar  oscillations  but  of  weaker  intensities 
will  be  set  up  in  circuits  1  and  2  if  they  are  adjusted  to  resonate 
to  3. 

If  small  inductances  and  capacitances  are  used  the  frequency 
of  the  oscillations  thus  produced  will  be  above  the  audible  range, 
and  special  means  for  detecting  them  must  be  employed.  A  con- 
venient method  is  to  use  a  pair  of  head  phones  and  a  crystal 
detector  such  as  is  commonly  employed  for  the  reception  of  radio 
signals.  Because  of  the  rectifying  action  of  the  point-crystal 
contact  the  high  frequency  alternating  voltage  across  the  con- 


148  ELECTRICITY  AND  MAGNETISM 

denser  will  produce  a  series  of  high  frequency  unidirectional 
pulses  in  the  phone  circuit.  Because  of  the  distributed  capaci- 
tance of  the  phone  windings,  these  are  smoothed  out  into  a  single 
pulse  which  causes  a  vibration  of  the  diaphragm.  The  sound  in 
the  phones  then  has  the  period  of  the  buzzer  armature.  If 
a  sufficient  amount  of  energy  is  available,  it  is  best  to  disconnect 
the  right-hand  phone  lead  shown  in  the  figure,  and  use  only  a 
single  wire  from  the  phone  through  the  crystal  to  the  oscillatory 
circuit.  This  is  particularly  important  when  the  condensers  are 
small  since  the  capacity  between  phone  leads  may  introduce  a 
very  appreciable  error. 

112.  Experiment  19.     Measurement  of  Inductance  and  Capaci- 
tance by  Resonance. — Connect  the  apparatus  as  shown  in  Fig.  81, 
using  for  LI  a  standard  inductance  variable  by  steps,  and  for  Ci 
a  variable  standard  air  condenser.     L2  should  be  a  single  layer 
coil  of  uniform  windings  whose  dimensions  may  easily  be  meas- 
ured.    Cz  should  be  an  air  condenser  with  plates  easily  accessible 
for  measurement.     First  obtain  resonance  in  circuit  2  by  varying 
the  frequency  of  the  source.    Next  obtain  resonance  in  circuit  1  and 
compute  the  LC  product.     Measure  the  dimensions  of  C2  and 
compute  its  capacity  from  eq.  (19)  given  on  page  94.     (See  also 
the  Appendix.)     Determine  LI  from  eq.  (60).     Check  your  result 
by  computing  the  inductance  of  LI  from  dimensions  using  the  for- 
mula given  in  the  Appendix. 

113.  Effective  Value  of  an  Alternating  Current. — If  an  alter- 
nating current  is  passed  through  an  ordinary  D.C.  ammeter,  no 
indication  will  be  registered,  since  such  an  instrument  indicates 
average  values,  which  in  this  case  is  zero.     However,  if  an  alter- 
nating current  is  passed  through  a  resistance,  heat  is  liberated, 
the  energy  of  which  is  furnished  by  the  current.     The  reason  for 
the  difference  in  effect  in  these  two  cases  is  that  the  torque  on  the 
moving  coil  of  the  instrument  is  proportional  to  the  current  and 
therefore  reverses  sign  with  it,  while  the  heating  effect  of  a  current 
is  proportional  to  its  square  and  is  therefore  positive  no  matter 
what  its  direction. 

It  is  customary  to  define  the  Effective  value  of  an  alter- 
nating current  as  the  equivalent  direct  current  which  liberates  the 
same  amount  of  heat  in  a  given  resistance  per  unit  time.  In 
deducing  the  relation  between  the  effective  value  of  an  alternating 
current  and  its  amplitude  or  maximum  value,  it  is  sufficient  to 
equate  the  heat,  in  joules,  developed  by  each  during  the  time  T 


ELEMENTARY  TRANSIENT  PHENOMENA  149 

of  one  complete  cycle.  Accordingly  let  i  =  /  sin  cot  be  the  alter- 
nating current  and  Ie  its  effective  value.  When  flowing  through 
a  resistance  R,  the  heat  liberated  by  each  is 


H  =  Ie*RT  =   C**Rdt  =  T*R    Hi 


tdt  (61) 


Therefore : 


Ie  =  ~  =  .707  /  (62) 

A/2 


The  above  process  is  seen  to  be  equivalent  to  squaring  the 
instantaneous  values  of  the  current,  taking  the  average  value 
of  the  squares,  and  then  extracting  the  square  root.  The 
effective  value  accordingly  is  often  spoken  of  as  the  "Root  Mean 
Square"  value.  The  same  considerations  hold  for  an  alter- 
nating E.M.F. 

114.  Power  Consumed  by  a  Circuit  Traversed  by  an  Alter- 
nating Current. — Let  us  suppose  that  an  alternating  E.M.F.  is 
impressed  upon  a  circuit  which  contains  reactance  as  well  as 
resistance  so  that  the  current  and  E.M.F.  are  not  in  phase.  It  is 
desired  to  find  the  power  consumed  by  the  circuit.  Let  the 
E.M.F.  and  current  be  given  respectively  by  the  following 
expressions 

e  =  E  sin  ut]  i  =  I  sin(o>£  ±  </>)  (63) 

where  </>  is  the  angle  of  lag  or  lead. 

The  energy  dH  consumed  in  the  time  dt  is 

dH  =  eidt  =  El  sin  ut  sin(co£  ±  <f>)dt  (64) 

The  energy  H  consumed  per  cycle  is 

CT. 
H  =  El  I  sin  cousin  ut  cos  <£  ±  cos  ut  sin  4>)dt 

{         CT  CT.  } 

=  El  \  cos  </>  I  sin2  wtdt  ±  sin  <f>  I  sin  ut  cos  utdl  [ 

I  Jo  Jo  ) 

=  El  {  cos  4>  I     (^ ^ —  }dt  ±  sin  <£  I  sin  coZ  cos  utdt  [ 

I          Jo    ™  *     '  Jo  } 


150  ELECTRICITY  AND  MAGNETISM 

Carrying  out  the  integrations  and  substituting  limits  the  last  two 
integrals  vanish  and  we  have 

H  =  El    J  cos  0 

z 

Energy  per  Cycle          E     I 

Power= 


It  is  thus  seen  that  the  power  consumed  is  the  product  of  the 
effective  E.M.F.  and  current  multiplied  by  the  cosine  of  the 
phase  angle.  Cosine  <f>  is  called  the  "  Power  Factor"  and  varies 
from  zero  to  unity. 


CHAPTER  XI 

SOURCES   OF  E.M.F.   ANEb  DETECTING  DEVICES  FOR 
BRIDGE  METHODS 

Before  discussing  the  various  bridges  which  are  to  be  employed 
in  the  measurement  of  inductance  and  capacitance,  the  student 
should  become  familiar  with  some  of  the  sources  of  alternating 
E.M.F.  and  detecting  devices  that  are  available.  Inasmuch  as 
the  alternating  currents  for  commercial  purposes  are  of  frequencies 
too  low  to  give  a  tone  suitable  for  telephonic  balances,  special 
generators  have  been  devised,  a  few  of  which  will  now  be  described. 

115.  The  Sechometer. — In  Exps.  12  and  17  methods  were 
employed  for  comparisons  of  capacitance  and  inductance  respect- 
ively in  which  batteries  were  employed  to  energize  the  bridges 
and  the  E.M.F. 's  due  to  the  reactances  were  made  manifest  during 
the  rise  and  fall  of  the  bridge  currents  following  the  closing  and 
opening  of  the  battery  circuit  key.  It  was  then  found  that  the 
galvanometer  deflected  in  one  direction  on  closing  and  in  the 
opposite  on  opening  this  key.  If  some  means  were  available  by 
which  the  galvanometer  leads  could  be  interchanged  each  time  the 
key  is  opened  and  closed,  the  deflections  would  always  be  in  the 
same  direction  and  if  the  interval  between  successive  operations 
of  the  key  were  small  compared  to  the  period  of  the  galvanometer, 
a  steady  deflection  would  result  whereby  the  sharpness  of  the 
bridge  balance  would  be  greatly  increased.  The  Sechometer 
is  a  device  which  accomplishes  this  purpose  and  derives  its  name 
from  the  "Secohm"  by  which  our  present  unit  of  inductance,  the 
henry,  formerly  was  known. 

It  consists  essentially  of  two  commutators  mounted  on  the 
same  shaft  which  may  be  driven  at  any  desired  speed  by  a  motor. 
The  segments  are  set  in  such  a  relation  to  each  other  that  the 
galvanometer  leads  are  interchanged  by  one  commutator  each 
time  the  polarity  of  the  battery  is  reversed  by  the  other.  The 
connections  are  shown  in  Fig.  82.  The  device  must  not  be  driven 
at  too  high  a  speed  since  sufficient  time  must  be  allowed  for  the 
establishment  of  a  steady  state  at  each  reversal.  High  speeds 

151 


152 


ELECTRICITY  AND  MAGNETISM 


also  develop  heat  at  the  brush  contacts  resulting  in  errors  due  to 
thermal  E.M.F.'s. 

It  is  well  to  get  an  approximate  bridge  balance  by  manipulating 


FIG.  82. — Sechometer  connections  to  bridge. 

the  battery  and  galvanometer  keys  with  the  sechometer  station- 
ary and  then  use  it  merely  to  obtain  the  final  setting.     The 


Fro.  83. — The  Sechometer. 

application  of  this  instrument  is  equivalent  to  using  a  generator 
giving  a  square  wave  form  and  can,  therefore,  be  used  only  with 
bridges  which  balance  independent  of  the  frequency.  Figure  83 


DETECTING  DEVICES 


153 


shows  the  assembled  instrument  provided  with  a  crank  for  hand 
driving. 

116.  The  Wire  Interrupter. — The  vibrating  wire  interrupter, 
shown  in  Fig.  84,  consists  essentially  of  a  piano  wire  stretched 
between  rigid  supports  A  and  B,  the  tension  of  which  may  be  varied 
by  the  screw  S.  Vibrations  are  maintained  by  means  of  an 
electromagnet  M,  iijtermitteMly  energized  by  a  battery  BI. 
The  mercury  cup  contact  C\  interrupts  this  current  when  the  wire 
is  drawn  up,  and  the  device  operates  in  a  manner  similar  to  the 
ordinary  buzzer.  The  battery  J32,  which  supplies  current  to  the 


FIG.  84. — Vibrating  wire. 

bridge,  is  connected  through  the  contact  €2  and  this  circuit  is 
also  closed  and  opened  at  the  frequency  maintained  by  the  wire. 

Frequencies  ranging  from  25  to  150  are  easily  secured.  This 
device  is  particularly  well  suited  for  use  with  the  vibration 
galvanometer,  since  it  permits  of  sharp,  easy  tuning,  and  may 
readily  be  adjusted  to  resonance  with  the  galvanometer.  Since 
the  vibration  galvanometer  responds  only  to  the  fundamental 
and  not  the  overtones,  the  fact  that  the  interrupter  gives  a  square 
wave  form  results  in  no  disadvantage  and  the  combination  may 
accordingly  be  used  on  bridge  circuits  which  do  not  balance 
independent  of  the  frequency  and  the  same  results  obtained  as 
though  a  source  giving  a  pure  sine  wave  were  employed.  If  a 
suitable  condenser  K  is  shunted  across  the  contact  Ci  to  prevent 
arcing,  the  device  will  operate  continuously  for  hours  with  little 
or  no  attention. 

117.  The  Motor  Generator. — Another  inexpensive  source  of 
alternating  current  is  the  small  motor  generator  set  manufactured 
by  the  Leeds  and  Northrup  Company  shown  in  Fig.  85.  The 
generator,  which  is  shown  at  the  right  in  the  figure,  is  of  the 


154 


ELECTRICITY  AND  MAGNETISM 


FIG.  85. — Motor  generator  set. 


inductor  type  and  has  stationary  windings  for  both  field  and 
armature  circuits.  Direct  current  is  supplied  to  the  field  coil  at 
the  base  thus  energizing  the  magnetic  circuit  which  includes  the 
broad  toothed  wheel  carried  on  the  armature  shaft  of  the  motor. 
The  reluctance  of  this  magnetic  circuit  depends  upon  the  position 

of  the  teeth  with  respect  to 
the  pole  pieces.  When  the 
wheel  is  driven,  the  flux 
through  the  magnetic  circuit 
fluctuates  at  a  frequency 
equal  to  the  number  of  teeth 
passing  the  pole  tips  per 
second.  An  alternating 
E.M.F.  is  thus  induced  in 
the  armature  windings  placed 
near  the  pole  tips. 
By  properly  choosing  the  shape  of  the  pole  tips  and  teeth,  it  is 
possible  to  obtain  a  wave  form  that  is  relatively  free  from  harmon- 
ics although  it  is  not  possible  to  eliminate  them  completely.  By 
the  use  of  a  suitable  filter,  good  wave  forms  may  be  obtained. 
By  means  of  a  specially  designed  mechanical  governor,  the  speed 
of  the  motor  may  be  maintained  constant  to  ^  per  cent. 

118.  The  Microphone  Hummer. — If  it  is  not  necessary  to 
supply  the  bridge  with  a  constant  frequency,  a  simple  microphone 
hummer  furnishes  a  convenient  and 
inexpensive  source  of  alternating  cur- 
rent. Such  a  circuit  is  shown  in  Fig.  86. 
It  consists  of  a  microphone  transmitter 
facing  a  telephone  receiver.  The  trans- 
mitter and  receiver  circuits  are  con- 
nected in  the  ordinary  way  by  a  tele- 
phone transformer.  Any  stray  sound 
will  cause  a  variation  in  the  microphone 
current  which  produces  a  sound  in  the 
receiver.  This  sound  is  "fed  back"  to 
the  microphone  which  again  produces  a 
sound  in  the  receiver  and  the  action  is 
thus  continuous,  the  energy  being  supplied  by  the  battery.  The 
interval  between  the  application  of  the  sound  to  the  microphone 
and  its  return  by  the  receiver  after  having  operated  the  electrical 
circuits  depends  to  a  large  extent  upon  the  time  constant  of  the 


FIG.  86. — Microphone 
hummer. 


DETECTING  DEVICES 


155 


secondary  or  receiver  circuit.  If  the  resistance  of  this  circuit  is 
not  too  large,  it  may  be  made  oscillatory  by  the  introduction  of  a 
condenser  as  indicated  in  the  diagram.  By  giving  suitable  values 
to  the  capacitance  of  this  condenser  a  large  range  of  frequencies 
may  be  obtained.  Another  trajisformer,  the  primary  of  which  is 
in  series  with  the  receiver,  furnishes  a  means  of  making  the  A.C. 
power  thus  generated  ^available' for  a  bridge  circuit.  A  conven- 
ient form  of  this  device  is  manufactured  by  R.  W.  Paul  of  London 
under  the  trade  name  "Kumagen,"  the  appropriateness  of  which 
is  easily  understood.  The  microphone,  receiver,  and  trans- 
formers are  contained  in  a  felt  lined  case  which  serves  to  deaden 
the  sound.  A  condenser  is  also  furnished  which  is  variable  in 
steps  chosen  so  as  to  give  a  number  of  suitable  frequencies. 
119.  The  Audio-oscillator. — The  frequency  of  the  microphone 
hummer,  described  above,  is  somewhat  variable  depending  upon 


FIG.  87. — Audio-oscillator. 

the  strength  of  the  driving  battery  and  the  load  upon 
the  secondary  of  the  output  transformer.  An  adaptation  of  the 
underlying  principle  has  been  made  by  Campbell  by  which  this 
objection  is  overcome.  It  consists  in  operating  the  microphone 
button,  not  by  sound  waves  from  a  telephone  receiver,  but  by 
means  of  a  tuning  fork  whose  mechanical  period  coincides  with 


156 


ELECTRICITY  AND  MAGNETISM 


the  period  of  the  electrical  circuit  which  it  energizes.  Several 
different  forms  are  on  the  market.  Figure  87  shows  an  instru- 
ment of  this  type  known  as  the  audio-oscillator,  manufactured  by 
the  General  Radio  Co.,  and  Fig.  88  gives  the  wiring  diagram. 
The  "field  coil"  which  is  connected  directly  across  the  battery 
serves  merely  to  magnetize  the  fork  and  armature  core  to  a  point 
on  the  magnetization  curve  near  the  maximum  permeability  and 
this  increases  the  attractive  forces  of  the  poles.  The  battery  also 
sends  current  through  the  microphone  and  primary  of  the  input 
transformer.  When  the  battery  key  is  closed,  the  current  through 


FIG.  88.^— Wiring  diagram  for  audio-oscillator. 

the  primary  of  the  input  transformer  induces  an  E.M.F.  in  the 
secondary  which  starts  oscillations  in  the  resonating  circuit 
which  includes,  besides  the  condenser  and  primary  of  the  output 
transformer,  the  armature  coil.  This  oscillating  current  changes 
the  attraction  between  the  armature  pole  tips  and  the  prongs  of 
the  fork.  Since  the  secondary  circuit  is  tuned  to  the  period  of  the 
fork,  the  fork  resonates  to  it,  thus  building  up  a  vigorous 
vibration.  The  microphone  button,  being  in  contact  with  the 
fork,  supplies  a  varying  current  of  this  same  frequency  to  the 
primary  of  the  input  transformer  and  energy  from  the  battery  is 
thus  furnished  to  maintain  the  oscillations,  and  carry  the  load  put 
upon  the  secondary  of  the  output  transformer. 

Each  transformer  coil  has  a  small  air  gap  to  prevent  distortion, 
but  their  magnetic  circuits  are  sufficiently  closed  to  prevent 
disturbing  stray  fields.  The  oscillator  is  self  starting  and  may 
be  placed  at  some  distance  and  operated  by  a  key  near  the  bridge. 
The  coils  are  so  wound  that  a  6-volt  battery  furnishes  ample 


DETECTING  DEVICES 


157 


power.  The  device  is  not  designed  to  furnish  more  power  than 
that  required  by  a  single  bridge  circuit.  If  overloaded,  the 
microphone  is  likely  to  pack.  It  is  carried  by  a  stiff  spring 
mounted  on  one  prong  of  the  fork  and  its  inertia  is  sufficient  to 
insure  response  to  vibrations  of  the  fork. 

120.  The  Vreeland  Oscillator. — None  of  the  sources  thus  far 
described  produce  alternating^  E.M.F.'s  of  a  purely  sinusoidal 
wave  form.  There  are  a  number  of  important  bridges  which  do 


FIG.  89. — Wiring  diagram  for  Vreeland  oscillator. 

not  balance  independent  of  the  frequency  and  when  a  telephone 
is  used  as  the  detecting  device,  complete  silence  can  not  be 
obtained  with  impure  wave  forms.  In  such  cases,  when  the 
fundamental  has  been  balanced  out,  the  overtones  are  still  heard 
and  materially  mar  the  sharpness  of  setting  which  would  other- 
wise be  possible. 

The  Vreeland  Oscillator  is  one  of  the  best  sources  of  pure  sine 
waves  available.  It  is,  in  reality,  a  mercury  arc  rectifier  operated 
backwards,  the  connections  for  which  are  shown  diagrammatically 
in  Fig.  89.  The  essential  part  of  the  device  is  a  large  pear  shaped 
mercury  arc  tube  with  two  anodes  AI  and  Az  having  a  common 


158  ELECTRICITY  AND  MAGNETISM 

mercury  cathode  KI.  It  is  well  known  that  the  mercury  arc  will 
operate  only  when  the  mercury  electrode  is  negative.  When  used 
as  a  rectifier,  the  condenser  and  deflecting  coil  are  removed  and 
the  source  of  alternating  E.M.F.  is  connected  to  the  terminals 
G.D.  When  G  is  positive  and  D  negative,  current  will  flow  from 
A  i  to  K i  through  the  battery  B,  which  is  here  shown  as  the  load, 
to  Z),  and  when  D  is  positive,  the  path  is  A2Ki  MG,  these  furnish- 
ing a  current  through  B  in  the  same  direction  as  before.  The 
reactances  Xi  and  X2  serve  to  smooth  out  the  fluctuations  through 
the  battery. 

To  understand  its  operation  as  an  oscillator,  let  us  suppose 
that  the  source  of  A.C.  is  removed  and  that  the  deflecting  coil  and 
condenser  are  connected  to  G  and  D  as  shown  in  the  figure.  The 
battery  B  now  becomes  the  source  of  power.  An  arc  is  started 
between  the  electrodes  KI  and  K2  by  shaking  the  tube  slightly, 
thus  causing  the  mercury  pools  to  unite  and  break  again.  The 
tube  is  quickly  filled  with  ionized  mercury  vapor  and  the  arc 
spreads  to  the  anodes  AI  and  A2.  The  switch  S  is  then  opened 
thus  stopping  the  arc  to  K2.  If  the  impedance  of  the  two  paths 
MD  and  MB  are  equal  and  the  tube  is  symmetrical,  the  arc  will 
divide  equally  between  the  anodes  A  i  and^4.2  which  are  thus  at  the 
same  potential  and  there  is  no  charge  in  the  condenser.  If, 
however,  some  irregularity  in  the  tube  causes  more  current  to  flow 
momentarily  to  the  anode  A  i  it  will  be  at  a  higher  potential  than 
A  2  and  a  charging  current  will  flow  to  the  condenser  through  the 
deflecting  coil  LL.  This  coil,  which  really  consists  of  two  parts, 
one  in  front  of  the  tube  and  the  other  behind  it,  is  placed  so  that 
its  magnetic  field  is  perpendicular  to  the  flow  through  the  arc. 
If  the  polarity  is  so  chosen  that  the  charging  current  deflects  the 
arc  stream  so  as  to  further  increase  the  current  to  A  i  a  very  appre- 
ciable charge  may  be  given  to  the  condenser.  When  the  condenser 
discharges,  the  deflecting  action  of  the  current  which  is  now 
reversed  will  cause  more  current  to  flow  to  the  anode  A2  thus 
raising  its  potential  above  AI  and  charging  the  condenser  in  the 
opposite  direction.  The  deflecting  coil  serves  the  double  purpose 
of  furnishing  a  self  inductance  to  form,  with  (7,  an  oscillatory  cir- 
cuit, and  to  automatically  deflect  the  arc  streams  from  one  anode 
to  the  other  to  maintain  the  oscillations.  The  frequency  is 
given  by  the  expression 


DETECTING  DEVICES  159 

where  L  is  the  inductance  of  the  deflecting  coil  in  henries,  and  C 
the  capacitance  of  the  condenser  in  farads.  It  is  found  that  such 
a  device,  when  properly  designed,  will  oscillate  at  frequencies 
ranging  from  100  to  4,000  cycles  per  second. 

Another  coil,  placed  near  the  deflecting  coils,  serves  as  the 
secondary  of  an  air  cored  transformer  to  supply  current  to  a 
bridge.  It  is  found  that  the  frequency  is  but  little  affected  by 
changes  in  the  load  on  the  secondary.  Because  of  the  relatively 
large  coils,  the  instrument  possesses  an  appreciable  stray  field 
and  must  be  placed  at  some  distance  from  the  bridge,  to  prevent 
direct  induction  in  the  coils  which  are  being  studied. 

121.  The  Electron  Tube  Oscillator. — One  of  the  simplest  and 
most  effective  means  of  obtaining  alternating  voltage  of  any 
desired  frequency  is  that  in  which  a  three  element  electron  tube 
is  used  to  maintain  continuous  oscillations  in  a  resonance  circuit. 
The  underlying  principle  is  the  amplifying  action  of  the  tube 
which  will  be  described  in  chap.  XIV.  It  will  be  sufficient  for 
the  present  purpose  to  point  out  that  the  electron  tube  consists 
of  a  highly  evacuated  glass  container  in  which  are  placed  a  fila- 
ment and  a  metal  plate  with  a  grid  mounted  between  them.  The 
grid  consists  of  a  fairly  coarse  meshed  structure  of  fine  wires. 
When  the  filament  is  heated  to  incandescence  by  an  electric 
current,  it  emits  electrons  which  may  be  drawn  to  the  plate  by 
a  battery  connected  through  an  external  circuit  between  the 
plate  and  filament.  The  positive  terminal  of  the  battery  must 
be  connected  to  the  plate. 

Inasmuch  as  the  electrons,  to  reach  the  plate,  must  pass  through 
the  meshes  of  the  grid,  the  number  arriving  at  the  plate  may  be 
controlled  by  giving  suitable  potentials  to  the  grid,  and  may  be 
stopped  entirely,  if  the  grid  is  sufficiently  negative.  Inasmuch 
as  the  energy  required  to  maintain  a  given  potential  on  the  grid 
is  small,  the  device  acts  as  an  electrical  throttle  valve,  whereby 
the  available  energy  of  the  plate  circuit  battery  may  readily 
be  controlled.  Figure  120  of  chap.  XIV  shows  the  relation  which 
exists  between  the  plate  current  and  grid  volts.  The  time 
required  for  an  electron  to  traverse  the  distance  from  filament 
to  plate  depends  upon  the  potential  of  the  plate  but  is  of  the 
order  of  10~8  seconds.  Changes  in  plate  current  accordingly 
follow  changes  in  grid  volts  with  remarkable  swiftness. 

There  are  many  different  circuits  in  which  an  electron  tube  may 
be  used  to  generate  sustained  oscillations.  Figure  90  shows  one 


160 


ELECTRICITY  AND  MAGNETISM 


of  the  simplest.  F,  G,  and  P  are  the  filament,  grid  and  plate 
respectively.  The  filament  is  heated  by  the  battery  A,  whose 
current  is  controlled  by  the  rheostat  R.  The  battery  B  furnishes 
the  potential  to  draw  the  electrons  from  the  filament  to  the 
plate.  The  inductance  L  and  the  condenser  C  form  the  oscilla- 
tory circuit.  To  understand  the  way  in  which  oscillations  are 
sustained,  let  us  suppose  that,  by  closing  the  switch  S,  the  estab- 
lishment of  a  current  through  the  coil  L  has  produced  a  transient 

oscillation  in  the  circuit  LC. 
s          |  This  would  quickly  die  out  if 

energy  were  not  supplied  to  it 
to  compensate  for  the  losses. 
Suppose  that  the  oscillatory 
current  through  L  is  in  the 
direction  of  the  arrow  and  is 
rising.  Due  to  the  self  in- 
ductance L  there  will  be  an 
E.M.F.  in  the  coil  in  the 
direction  DE.  This  lowers 
the  potential  of  the  grid  with 
respect  to  the  filament  which 
thus  decreases  the  plate  cur- 
rent, flowing  through  the 
part  Lp.  This  decrease  in  the 

an 
A 


FIG.  90. — Electron  tube  oscillator. 


plate  current  induces  in  Lp 

E.M.F.  which  tends  to  keep  the  oscillatory  current  flowing, 
continuation  of  this  reasoning  throughout  the  changes  occurring 
during  a  complete  cycle  will  show  that  the  variations  in  plate 
current  always  induce  in  Lp  an  E.M.F.  tending  to  drive  the 
oscillatory  current  in  L  in  the  direction  in  which  it  happens 
to  be  flowing  at  any  instant.  The  oscillations  would  increase 
indefinitely  in  amplitude  were  it  not  for  the  fact  that  the  grid 
volt-plate  current  characteristic  of  the  tube  becomes  horizontal 
at  each  end. 

Frequencies  ranging  from  1  cycle  to  several  millions  per 
second  may  be  obtained.  Alternating  current  power  for  bridge 
work  may  be  obtained  either  by  placing  another  coil  near  L 
which  then  serves  as  the  secondary  of  an  air  core  transformer  or 
by  connecting  the  primary  of  a  telephone  transformer  in  the 
plate  circuit  as  shown  in  the  figure.  The  latter  is  to  be  preferred 
since  variations  in  the  load  have  a  smaller  disturbing  effect  upon 


DETECTING  DEVICES  161 

the  frequency  than  is  the  case  with  the  former  arrangement.  The 
wave  form  is  not  as  free  from  harmonics  as  that  obtained  from  a 
Vreeland  oscillator,  and  a  filter  must  be  used  in  cases  where 
extreme  purity  is  essential. 

DETECTING  DEVICES 

122.  Telephone  Receiver.1 — The  telephone  receiver  is  one  of 
the  most  generally  useful  of  the  various  instruments  for  detecting 
the  balance  condition  in  a  bridge  circuit  actuated  by  alternating 
currents.  It  consists  essentially  of  a  horseshoe  magnet  upon 
which  is  wound  a  pair  of  coils  carrying  the 
current  to  be  detected,  and  a  soft  iron  dia- 
phragm mounted  near  the  poles  as  shown  in 
Fig.  91.  The  current  through  the  coils  mag- 
netizes the  core  which  attracts  the  diaphragm 
with  a  force  proportional  to  the  square  of  the 
induction  produced.  The  sensitivity  of  the 
receiver  is  increased  by  using  for  the  core, 
not  a  piece  of  soft  iron,  but  a  permanent 
magnet.  The  way  in  which  this  is  brought  D 

i  i  f  ,i       f   -it  FIG.  91. — Telephone 

about  may  be  seen  from  the  following  con-  receiver, 

sideration.     Let  B0  be  the  constant  induction 
through  the  gap  due  to  the  permanent  magnet,  and  let  the  addi- 
tional induction  which  is  proportional  to  the  current  i  in  the 
coils  be  kii.     Then  the  total  pull  on  the  diaphragm  is  given  by 

"Pi-ill    r,    r>2    T.  ( Z?     _1_  j%  ^2  !•    R  2  _l_  OJ.  £•    R  V  _|_  1-  2J.  V2 

JrU.ll    —    /t/2-O        —   /V2\-OO    "I     ">Iv}        —    n/2J->Q     ~\      At\/i_lv2J-)Ql'     \^  A/1   n/^l/ 

The  first  term  represents  the  pull  due  to  the  permanent  magnet 
alone,  the  second,  that  due  to  the  current  and  magnet  combined, 
while  the  third  is  that  due  to  the  current  alone.  If  it  is  desired  to 
have  the  motion  of  the  diaphragm  follow  the  variations  in  the 
current  so  that  its  motions  may  reproduce  for  the  human  ear 
the  sound  waves  acting  upon  the  diaphragm  of  a  distant  tele- 
phone transmitter,  then  the  receiver  must  be  so  designed  that  the 
second  term  is  large  compared  to  the  last  which  contains  the 
square  of  the  current.  The  first  term  need  not  be  considered 
since  it  is  independent  of  the  current.  The  desired  effect  is 
attained  by  making  BQ  large  compared  to  kii.  Since  BQ  enters 
as  a  factor  in  the  second  term,  making  it  large  has  the  effect  of 

1  MILLS,  Radio  Communication,  p.  27. 
11 


162 


ELECTRICITY  AND  MAGNETISM 


increasing  the  motion  of  the  diaphragm  and  hence  of  making  the 
receiver,  to  a  certain  extent,  an  amplifying  device. 

If  the  third  term  is  not  negligible  compared  to  the  second, 
then,  although  there  is  a  repetition  with  amplification  there  is 
also  distortion  since  the  pull  which  it  defines  is  proportional  to  the 
square  of  the  current.  The  nature  of  this  distortion  can  be 
understood  by  supposing  that  the  current  is  sinusoidal,  e.g., 
i  =  I  sin  cot.  The  last  term  then  becomes 


sin2  co*  = 


I  —  cos 


Quartz 
Fibre 


Mirror 


and  it  is  seen  that  the  distorting  pull  is  made  up  of  two  parts: 

kjci2!2 
A  constant  part  — ^ —  which  need  not  be  considered  and  a 

pulsating  part  having  twice  the  frequency  of  the  phone  current. 
Since  the  diaphragm  of  the  telephone  receiver  is  an  elastic 
body  it  will  have  a  frequency  of  its  own  and  will  accordingly 
KN  respond  more  vigorously  to  frequencies  which 

correspond  to  its  natural  period,  and  another 
source  of  distortion  is  thus  introduced.  For 
bridge  work,  however,  this  fact  may  be  utilized 
to  increase  the  sensitivity  by  impressing  upon 
the  bridge  the  frequency  to  which  the  tele- 
phone resonates.  Phones  for  this  particular 
purpose  are  constructed  in  such  a  way  that 
their  resonance  frequencies  may  be  varied 
over  a  considerable  range. 

123.  Thermo-galvanometer. — The  Duddell 
Thermo-galvanometer  is  an  adaptation  of  the 
Boys'  radio-micrometer  for  the  purpose  of 
measuring  and  detecting  small  alternating 
currents.  The  moving  system,  shown  in  Fig. 
92,  consists  of  a  single  turn  of  silver  wire  at 
the  bottom  of  which  is  a  tiny  thermocouple 
of  bismuth  and  antimony.  The  system  is 
suspended  by  means  of  a  fine  quartz  fibre  between  the  poles  of  a 
strong  horseshoe  magnet  and  carries  a  small  mirror  by  means  of 
which  its  deflections  are  read  with  a  lamp  and  scale.  Imme- 
diately below  the  thermo-j  unction  is  mounted  a  resistance  unit 
through  which  the  current  to  be  measured  is  passed.  The  heat 
from  this  current  is  carried  to  the  thermo-j  unction  by  convection 


Bi 


i  rr\si 


rr 


DETECTING  DEVICES  163 

and  radiation  and  causes  a  current  to  flow  through  the  low 
resistance  silver  loop  which  is  deflected  by  the  electrodynamic 
action  of  the  field.  Since  the  heating  effect  is  proportional  to 
the  square  of  the  current  while  the  thermal  E.M.F.,  for  small 
temperature  differences,  is  proportional  to  the  temperature,  the 
indications  of  this  instrument  are  roughly  proportional  to  the 
square  of  the  current.  ^ 

Several  heating  umts  are  provided  with  each  instrument  and 
range  in  value  from  1  to  1,000  ohms  according  to  the  current 
sensitivity  desired.  For  low  resistances,  they  are  made  of  fine 
wire  bent  back  and  forth  but  for  the  higher  values,  fine  platinized 
quartz  fibres  are  used.  With  the  latter,  current  sensitivities  of 
10~5  amperes  are  obtained.  This  type  of  instrument  may  be 
calibrated  on  direct  currents  and  then  used  to  measure  alter- 
nating currents.  Since  it  is  practically  free  from  inductance, 
the  instrument  may  be  used  for  the  measurement  of  currents  of 
very  high  frequencies.  Because  of  the  low  resistance  of  the 
moving  system  it  is  critically  damped  electromagnetically,  and  is 
usually  designed  so  as  to  have  a  period  of  from  three  to  four 
seconds.  Because  of  the  delicacy  of  the  quartz  fibre  suspension 
and  the  light  silver  loop,  it  is  not  a  robust  instrument  and  must  be 
handled  with  caution.  The  heating  elements  are  easily  burned 
out  and  should  always  be  protected  by  a  high  resistance  which 
may  be  reduced  to  zero  when  it  has  been  ascertained  that  safe 
limits  of  current  will  not  be  exceeded.  Sudden  changes  in  tem- 
perature cause  the  zero  to  drift  and  the  instrument  is  usually 
enclosed  in  a  tight  wooden  case. 

124.  Vibration  Galvanometer.1 — The  vibration  galvanometer 
is  one  of  the  most  useful  instruments  available  for  the  detection  of 
minute  alternating  currents  of  commercial  frequencies.  To 
secure  suitable  sensitivity,  advantage  is  taken  of  the  principle  of 
resonance.  That  is,  the  moving  system  is  so  adjusted  mechan- 
ically, that  its  natural  period  coincides  with  that  of  the  alter- 
nating current  to  be  detected.  Although  the  instrument  shows 
very  little  response  to  direct  currents  or  to  alternating  currents 
to  which  it  is  not  tuned,  nevertheless  when  resonance  has  been 
secured,  a  very  appreciable  vibration  results.  The  vibrations 
are  indicated  by  means  of  a  small  concave  mirror  carried  on  the 
moving  system  which  focuses  the  image  of  an  incandescent  fila- 

1  LAWS,  Electrical  Measurements,  p.  434. 
WENNER,  Bull.  U.  S.  Bureau  of  Standards,  vol.  6,  1909-10,  p.  347. 


164 


ELECTRICITY  AND  MAGNETISM 


ment  on  a  ground  glass  scale.  When  the  system  vibrates, 
the  image  is  drawn  out  into  a  broad  band  of  light,  while  very 
slight  motions  are  detected  by  a  diminution  in  the  sharpness  of 
the  line. 

One  of  the  chief  reasons  for  the  superiority  of  this  instrument 
is  the  fact  that  its  response  is  selective.  In  many  measurements 
it  is  necessary  to  use  a  pure  sine  wave,  a  thing  difficult  to  secure. 
Since  vibration  galvanometers  may  be  made  with  a  selectivity 


—SPRING 


FIG.  93.— Leeds  and  Northrup 
vibration  galvanometer. 


FIG.  94. — Tuning  mechanism  for  Leeds 
and  Northrup  vibration  galvanometer. 


so  high  that  their  response  to  the  third  harmonic  is 


1 


to  the  fundamental  and  to  the  fifth, 


1 


4,000 


of  that 


impure  waves  may 


be  employed  with  very  little  if  any  inaccuracy  introduced.  In 
fact  an  interrupter  of  the  vibrating  wire  type  described  above, 
giving  a  square  wave  form,  may  be  employed.  The  current 
sensitivity  of  the  vibration  galvanometer  is  about  the  same  as 
that  of  a  good  telephone  receiver  —  i.e.,  10~6  amperes. 

Obviously  the  instrument  may  be  of  either  the  D'Arsonval  or 
the  Thomson  type.  In  Fig.  93  is  shown  one  of  the  former,  or 
moving  coil  instruments,  while  Fig.  94  shows  how  the  moving 


DETECTING  DEVICES  165 

system  is  tuned.  The  coil  is  held  in  position  by  a  taut  phosphor- 
bronze  ribbon,  the  effective  length  of  which  is  varied  by  means  of 
the  movable  bridge  carried  on  the  upper  screw.  By  sliding  this 
bridge  up  or  down  rough  tuning  is  obtained  while  fine  adjust- 
ments are  secured  by  slightly  changing  the  tension  of  the  sus- 
pension by  means  of  the  lower  'screw  and  spring. 

Figure  95  shows  a  Tinsley  instrument  of  the  moving  magnet 
type.  The  vibrating^  system  consists  of  a  small  permanent 
magnet  mounted  on  a  taut  metallic  ribbon  behind  which  is  held 


FIG.  95. — Tinsley  vibration  galvanometer. 

the  fixed  deflecting  coil.  Specially  shaped  pole  pieces  concen- 
trate the  field  of  the  large  horseshoe  magnet  on  the  moving 
magnet.  Since  the  period  of  the  system  is  determined  largely  by 
the  strength  of  this  external  field,  tuning  is  obtained  by  changing 
this  field.  This  is  accomplished  by  moving  the  soft  iron  magnetic 
shunt  along  the  horseshoe  magnet.  The  milled  head  shown  at 
the  front  of  the  base  operates  a  worm  gear  which  moves  the  shunt. 
125.  Alternating  Current  Galvanometer. — The  alternating 
current  galvanometer  is  one  of  the  most  sensitive  devices  avail- 
able for  detecting  the  balance  condition  in  a  bridge  supplied  with 
an  alternating  E.M.F.  It  is  essentially  a  D'Arsonval  galva- 
nometer with  the  permanent  magnet  replaced  by  an  electro- 


166 


ELECTRICITY  AND  MAGNETISM 


A.C. 

Supply 


To  Bridge 


magnet  energized  from  the  same  A.C.  source  as  that  supplying 

the  bridge.     It  operates  upon  the  electrodynamometer  principle 

and  the  direction  of  the  torque 
acting  upon  the  moving  coil  is 
independent  of  the  polarity  of 
the  supply.  Its  operation  is 
complicated  by  the  fact  that, 
when  connected  to  the  bridge, 
there  are  present  in  the  coil 
two  currents,  one  due  to  the 
unbalanced  condition  of  the 
bridge,  and  one  induced  by 
the  alternating  flux  of  the 
galvanometer  field.  Since  the 
former  is  small  and  disappears 
at  balance,  the  latter  by  far 
overpowers  it  and  must  either 

be  eliminated  or  made  ineffective. 

It  may  be  shown  in  the  following  manner  that  when  the  current 

induced  in  the  coil  is  90°  out  of  phase  with  the  flux  through  it, 

the  torque  is  zero. 

Let  <f>  =  <l>sinco£  =  instantaneous   flux    through   the    coil  and 

i  —  Ism(a)t  ±  0)    instantaneous    current   in   the  coil. 

The  torque  acting  upon  the  coil  in  a  given  position  at  any 

instant  is  then 

T  =  K&  sin  ut  I  sin  (orf  ±  0) 

where  K  is  a  constant  depending  upon  the  geometry  of  the  instru- 
ment. The  average  value  r  taken  over  the  time  T  of  one  complete 
cycle  is 

T  . 
sin  co£  sm(co!  +  0)dt 


FIG.  96. — Alternating  current 
galvanometer. 


T    = 


*¥Tsi 


sin  cousin  ut  cos  0  ±  cos  ut  sin  0)dt 


cos 


JT  .  CT  1 

sin2  utdt  ±  sin  0  I     sin  ut  cos  utdt 

T  1  -  cos  2wt  ,        sin  0  sin2  «H  T 

-dt  +  -  = 

2  2         J0 


cos  0 


The  resultant  torque  on  the  coil  is,  accordingly,  positive  or 


DETECTING  DEVICES  167 

negative  depending  upon  the  sign  of  0  and  is  zero  for  6  =  +-• 

2 

Since  the  E.M.F.  induced  in  the  coil  is  90°  out  of  phase  with  the 
flux  producing  it,  the  condition  stated  above  is  equivalent  to 
saying  that  the  induced  current  must  be  in  phase  with  the  E.M.F. 
In  other  words,  the  bridge  circuit  to  which  the  coil  is  connected 
must  be  nonreactive.  In  certain  bridges  such  as  those  for  com- 
paring two  condensers  or  two  inductances  this  is  obviously 
impossible.  The  required  condition  may,  however,  be  met  by 
shunting  across  the  coil  an  appropriate  variable  reactance,  e.g., 
an  inductance  with  a  variable  series  resistance  in  the  former  case, 
or  a  condenser  and  resistance  in  the  latter. 

When  the  galvanometer  is  first  connected  to  the  bridge,  it  will 
be  found  that,  due  to  the  action  just  described,  the  coil  assumes  a 
very  rigid  position,  including  either  a  maximum  or  a  •minimum 
amount  of  the  field  flux.  If  the  former  position  results,  a  leading 
current  through  the  coil  is  indicated,  and  a  shunt  with  an  induc- 
tive reactance  must  be  applied.  For  satisfactory  operation,  a 
certain  amount  of  stability  is  required  to  give  a  constant  zero 
position,  so  inductive  reactance  across  the  coil  should  predomi- 
nate. Since  the  reactance  of  the  bridge  is  an  important  factor  in 
determining  the  rest  position  of  the  coil,  the  galvanometer  key 
must  remain  closed,  and  the  balance  established  by  opening  and 
closing  the  supply  circuit  to  the  bridge. 

The  alternating  current  galvanometer  has  an  important  advan- 
tage over  detecting  devices  such  as  the  telephone  or  vibration 
galvanometer,  in  that  it  swings  to  the  right  or  left  according  to 
the  phase  of  the  current  at  the  galvanometer  corners  of  the  bridge 
while  with  the  latter,  no  such  effect  is  possible.  Furthermore, 
if  a  direct  current  is  supplied  to  the  field,  it  becomes  an  ordinary 
D'Arsonval  galvanometer  and  may  be  used  to  determine  the 
steady  state  balance.  Its  sensitivity  may  be  made  100  times  that 
of  the  telephone  or  vibration  galvanometer.  It  has,  however, 
one  distinct  disadvantage,  in  that  the  deflection  depends  not 
only  upon  the  field  and  current  through  the  coil  but  also  upon  the 
phase  angle  between  them.  It  can  not  therefore  be  calibrated  to 
measure  currents.  Furthermore,  zero  deflection  indicates  either 
no  current,  or  current  90°  out  of  phase  with  the  field.  A  simple 
test  for  the  latter  condition  is  to  shift  the  phase  of  the  field  by 
inserting  a  resistance  in  series  with  the  field  coil. 


CHAPTER  XII 
ALTERNATING  CURRENT  BRIDGES 

126.  General  Considerations. — In  order  to  obtain  the  reactive 
effect  of  an  inductance  or  a  capacitance  it  is  necessary  that  the 
current  through  it  should  be  variable.  In  the  early  bridge  meas- 
urements for  comparing  inductances  or  capacitances  and  even 
for  determining  an  inductance  in  terms  of  a  capacitance  the  vari- 
able current  was  obtained  simply  by  closing  and  opening  the 
battery  circuit  leaving  the  galvanometer  permanently  connected 
to  the  bridge.  The  galvanometer  employed  was  usually  of  the 
long  period  ballistic  type.  This  procedure  is  open  to  two  objec- 
tions. First,  the  sensitivity  thus  realizable  is  not  great  and 
second  it  may  lead  to  results  which  are  appreciably  different  from 
the  effective  values  of  the  condensers  or  coils  when  employed, 
as  is  usually  the  case,  in  circuits  traversed  by  alternating 
currents.  For  example,  the  effective  value  of  the  self  in- 
ductance of  the  primary  of  a  transformer  when  an  alternating 
current  is  flowing  through  it,  depends  upon  the  load  across 
the  secondary.  If  measured  by  the  make  and  break  method 
with  a  ballistic  galvanometer  as  the  detecting  device,  the  result 
is  the  inductance  of  the  primary  alone  independent  of  the  effect 
of  the  secondary. 

The  ballistic  galvanometer  is  an  integrating  instrument,  and  a 
zero  deflection  does  not  necessarily  mean  that  no  current  has 
passed  through  it,  but  that  equal  and  opposite  quantities  have 
traversed  it.  The  bridge  may  have  been  out  of  balance  each  way 
during  the  time  the  current  through  it  was  changing.  It  is, 
accordingly,  much  better  to  use  alternating  currents  through 
the  bridge  and  employ  a  detecting  device  such  as  the  telephone  or 
vibration  galvanometer,  a  zero  indication  of  which  indicates  that 
at  no  time  is  there  a  current  through  it,  and  that  the  bridge  is 
balanced  at  all  times. 

It  will  appear  in  the  discussion  which  follows  that,  in  order  for  a 

168 


ALTERNATING  CURRENT  BRIDGES  169 

bridge  with  reactive  members  to  be  balanced  at  all  times,  there 
are  two  conditions  which  must  be  satisfied.  First,  the  bridge 
must  be  balanced  for  direct  currents,  " steady  state  balance," 
and  second,  it  must  be  balanced  for  alternating  currents,  "vari- 
able state  balance."  These  two  balance  conditions  may  be 
interpreted  in  the  equation  for  the  bridge  in  a  simple  manner. 
An  expression  is  deduced,  involving  one  current  and  its  time 
derivative.  The  "steady  state  balance"  means  that  the  coeffi- 
cient of  the  current  term  is  zero.  The  "variable  state  balance" 
means  that  the  coefficient  of  the  term  for  the  changing  current, 
i.e.,  the  time  derivative,  is  zero.  This  applies  to  any  bridge 
for  which  the  balance  condition  may  be  reduced  to  an  ex- 
pression involving  only  one  current  and  its  first  time  deriva- 
tive. Such  a  bridge  balances  independent  of  the  frequency. 
If  a  second  time  derivative  is  involved,  as,  for  example,  in 
Trowbridge's  bridge  and  the  frequency  bridge,  the  wave  form 
of  the  current  must  be  assumed,  and  the  bridge  no  longer 
balances  independent  of  the  frequency.  In  some  instances,  the 
student  will  find  it  advantageous  to  obtain  the  former  by  the 
use  of  a  battery  and  direct  current  galvanometer,  and  later 
apply  an  alternator  and  detector  for  the  variable  state  balance. 
After  becoming  experienced  in  this  type  of  work,  however, 
both  balances  may  be  obtained  simultaneously  by  the  use  of 
alternating  currents. 

127.  Maxwell's  Bridge.1 — One  of  the  simplest  methods  for 
determining  an  inductance  in  terms  of  a  capacitance  or  vice 
versa  is  the  method  known  as  Maxwell's  bridge.  It  consists  of 
an  ordinary  Wheatstone's  bridge  with  three  non-inductive 
resistances  RI,  R2,  and  J?3,  as  shown  in  Fig.  97,  while  the  fourth 
arm  contains  the  inductance  L  to  be  determined.  Let  the  ohmic 
resistance  of  this  coil  be  /?4.  To  offset  the  reactance  of  the  coil 
L,  a  condenser  C  is  placed  across  the  opposite  resistance  RI. 
When  an  alternating  E.M.F.  is  applied  to  the  bridge,  the 
current  in  the  upper  half  will  lead  the  E.M.F.,  while  that 
in  the  lower  half  lags  behind  it.  Accordingly,  if  an  A.C. 
galvanometer  or  other  detecting  device  is  connected  ahead  of 
the  inductance  and  behind  the  condenser,  the  arms  of  the  bridge 
may  be  so  adjusted  that  the  potential  changes  at  D  and  E  are 

1  MAXWELL*  Electricity  and  Magnetism,  vol.  2,  p.  387. 


170 


ELECTRICITY  AND  MAGNETISM 


not  only  equal  but  also  in  phase,  and  no  indication  of  the  instru- 
ment will  result. 

The  conditions  necessary  for  balance  may  be  obtained  in  the 
following  manner.  Let  the  instantaneous  currents  through  the 
various  elements  be  designated  as  in  the  figure.  By  equating 


FIG.  97. — Maxwell's  bridge. 


the  fall  of  potential  from  A  to  D  to  that  from  A  to  E,  and  the  fall 
from  D  to  B  to  that  from  E  to  B  and  noting  that  i2  is  made  up 
of  i  and  ii  the  following  equations  result. 

t'«  =  *i  +  i  (1) 

R\i\  —  R&s  (2) 

7?    •               Z?     '              7   ^3  /0x 

«jtj   —   «4t|  T  **  "S  V^J 


*1  =  ^  I  idt 


(4) 


We  thus  obtain  four  equations  between  the  four  currents.  The 
currents  may  therefore  be  eliminated  and  the  relations  between 
L,  C,  and  the  R's  obtained  which  are  necessary  for  a  balance. 
Eliminating  iz  between  eqs.  (1)  and  (3)  there  results 


(5) 


ALTERNATING  CURRENT  BRIDGES  171 

Differentiating  eq.   (4)  with  respect  to  t  and  solving  for  i,  also 
substituting  the  value  of  is  from  eq.  (2)  in  eq.  (5),  we  have 


'  r>  D 

If  the  bridge  has  first  been  balanced  for  the  steady  state,     * 

=  Rz,  whence  only  the  terms  containing  the  derivative  of  ii 
remain.  The  second  condition  for  balance  is  obtained  by  equat- 
ing the  coefficients  of  the  derivatives,  whence 

L  =  RiR3C  (7) 

While  the  theory  of  this  bridge  is  simple,  its  application  in  the 
laboratory  is  somewhat  tedious  in  case  both  L  and  C  are  fixed. 
For  example,  suppose  a  steady  state  balance  has  been  obtained, 
and  it  is  attempted  to  satisfy  eq.  (7)  by  changing  R%  or  Rs.  The 
steady  state  balance  is  immediately  upset  and  must  again  be 
obtained  before  the  test  for  the  new  value  of  R2  or  Rs  can  be  made. 
If  L  or  C  are  continuously  variable,  eq.  (7)  may  be  satisfied  with- 
out disturbing  the  steady  state  balance,  and  it  is  in  this  case  that 
the  bridge  is  particularly  useful.  An  experienced  observer 
however,  quickly  learns  to  make  both  balances  simultaneously. 

128.  Experiment  20.  Maxwell's  Bridge  for  Self  Inductance. — 
Make  the  connections  as  shown  in  Fig.  97  using  for  L  a  continu- 
ously variable  inductance  and  for  C  a  subdivided  condenser. 
As  a  source  of  E.M.F.,  use  an  alternator  giving  a  frequency  from 
500  to  1,000  cycles  per  second,  and  a  pair  of  head  phones  as  the 
detector.  It  may  be  well  to  use  a  battery  and  ordinary  galvanom- 
eter to  obtain  the  steady  state  balance.  Connect  double  pole 
double  throw  switches  so  that  each  source  and  detector  may  be 
quickly  exchanged.  •  For  the  steady  state  balance,  care  must  be 
taken  to  close  the  battery  key  before  the  galvanometer  key. 
Obtain  the  variable  state  balance  by  changing  L.  If  the  balance 
does  not  lie  within  the  range  of  L,  change  either  C  or  one  of  the 
resistances  of  eq.  (7).  If  the  latter  is  done,  a  new  steady  state 
balance  must  be  obtained. 

Report. — Plot  a  calibration  curve  of  L  as  a  function  of  its  scale 
readings.  Define  coefficient  of  self  inductance.  If  a  copper 
disk  were  held  near  the  coil  so  that  its  face  is  perpendicular  to  the 
axis  of  the  coil  would  the  inductance  as  measured  in  this  manner 
be  changed?  Explain. 


172 


ELECTRICITY  AND  MAGNETISM 


129.  Anderson's    Modification    of    Maxwell's    Bridge.1 — It 

was  pointed  out  above  that  the  adjustments  for  balancing  Max- 
well's bridge  are  likely  to  be  tedious  because  each  attempt  to 
obtain  a  variable  state  balance  necessitates  a  redetermination  of 
the  steady  state  balance.  Anderson  has  suggested  a  simple 
device  by  which  the  variable  state  balance  may  be  obtained 
without  destroying  that  for  steady  states.  The  connections  are 
shown  in  Fig.  98.  It  will  be  noted  that  the  condenser  C,  instead 
of  being  connected  to  the  point  D  has  the  resistance  r,  placed  in 


FIG.  98. — Anderson's  modification  of  Maxwell's  bridge. 

series  with  it  so  that  its  time  constant  may  be  varied.  Since,  in 
determining  the  steady  state  balance,  the  condenser  produces  no 
effect,  it  may  be  left  in  circuit  during  that  process,  and  the  only 
change  introduced  is  placing  r  in  series  with  the  galvanometer. 
This  reduces  slightly  the  sharpness  of  balance  which  is  of  little 
consequence.  The  steady  state  balance  may  accordingly  be 
made  once  for  all,  and  the  variable  state  balance  obtained  by 
adjusting  r  to  the  proper  value. 

The  determination  of  the  balance  condition  is  somewhat  more 
complicated  and  is  as  follows:  Let  the  instantaneous  currents 
through  the  various  elements  of  the  bridge  be  designated  as 
before.  The  points  P  and  E  are  now  the  ones  remaining  at  the 
same  potential.  Accordingly, 

t2  =  ii  +  i  (1) 


if 

«/ 


-  I  idt  +  ri 


(2) 


Phil.  Mag.,  vol.  31,  1891,  p.  329. 


ALTERNATING  CURRENT  BRIDGES 


173 


it  +  ri  =  L-j.  +  #4^3 
Combining  eqs.  (1)  and  (3)  with  eq.  (4),  there  results 


(3) 

(4) 

(5) 
(6) 


Imposing  now  the  condition  for  the  steady  state  balance,  it  is 
seen  that  the  coefficients  of  the  integrals  are  equal  and  eq.  (6) 
then  becomes 

Rtf   ,  L  /r.N 


Eliminating  ii  between  eqs.  (2)  and  (5)  we  have 


Rearranging  and  using 


(8) 


FIG.  99. — Stroude  and  Gates  bridge. 

Another  change  in  the  arrangement  of  this  bridge  has  been 
suggested  by  Stroude  and  Gates1  which  is  usually  an  advantage. 
The  general  theory  of  bridges  shows  that  it  is  always  possible  to 
interchange  the  source  of  power  and  the  detecting  device.  Figure 
99  shows  the  connections  when  this  has  been  done  with  a  slight 

1  Phil  Mag.,  vol.  6,  1903,  p.  707. 


174  ELECTRICITY  AND  MAGNETISM 

change  in  the  arrangement  which  improves  the  ease  of  manipula- 
tion. The  principal  advantage  in  this  method  lies  in  the  fact 
that  r  is  now  in  series  with  the  bridge  and  a  correspondingly 
higher  E.M.F.  may  be  used  without  injuring  the  resistances.  An 
increase  in  sensitivity  is  thus  secured.  The  same  balance  condi- 
tion, eq.  (8),  applies. 

130.  Experiment    21.     Stroude    and    Oates    Bridge   for    Self 
Inductance. — Connect  the  apparatus  as  shown  in  Fig.  99.     For 
power  supply  and  detector  use  either  an  audio  frequency  gen- 
erator and  phones  or  city  A.C.  supply  and  alternating  current 
galvanometer.     The  latter  is  particularly  well  adapted  to  this 
bridge.     Arrange  double  pole  double  throw  switches  so  that  a 
direct  current  source  and  ordinary  galvanometer  may  quickly  be 
substituted  for  making  the  steady  state  balance. 

As  an  unknown  inductance,  use  two  coils  mounted  in  a  fixed 
position  close  enough  to  one  another  so  that  mutual  inductance 
exists  between  them.  Measure  the  inductance  of  each  sepa- 
rately, then  connect  them  in  series  and  measure  the  resultant 
inductance  with  the  connections  direct  and  reversed,  that  is  with 
the  mutual  inductance  first  aiding  and  then  opposing  the  self 
inductances.  Calling  LI  and  L2  the  self  inductances  of  the 
individual  coils,  and  La  and  L0  the  two  together  when  aiding  and 
opposing  respectively,  the  following  equations  hold 

La  =  Li  +  L2  +  2M  (9) 

Lo  =  L!  +  L2  -  2M 

Report. — Check  your  results  by  solving  eq.  (9)  for  M.  Give 
a  physical  interpretation  for  eq.  9. 

131.  Trowbridge's  Method  for  Self  Inductance.— In  Art.  141 
there  will  be  described  a  method  by  which  an  inductance  may  be 
measured  in  terms  of  capacitance  using  the  two  reactances  in 
series  in  one  arm  of  a  bridge.     While  this  arrangement  admits  of 
an  exceedingly  sharp  adjustment,  the  bridge  may  be  balanced  for 
only  one  frequency  for  given  values  of  L  and  C.     In  fact  one  of  its 
most  useful  applications  is  the  determination  of  frequency  using 
reactances  of  known  magnitudes.     Trowbridge1  has  shown  that 
if  the  reactances  are  shunted  with  properly  chosen  resistances  the 
balance  condition  may  be  made  independent  of  the  frequency 
while  sensitiveness  of  balance  is  very  inappreciably  sacrificed. 
Such  an  arrangement  is  shown  in  Fig.  100. 

1  Phys.  Rev.,  vol.  23,  1905,  p.  475. 


ALTERNATING  CURRENT  BRIDGES 


175 


Let  the  currents  be  designated  as  indicated  in  the  figure.  Then 
putting,  for  the  moment,  R*  =  0,  the  following  equations  must 
hold  for  the  balance  condition : 


~ 


*•  =*  T*  I  isdt 


i,  =  i,  +  i, 
For  simplicity  let  R\  =  R%,  then  i\  —  iz  . 


(1) 
(2) 

(3) 

(4) 

(5) 
(6) 


FIG.   100. — Trowbridge's  method  for  self  inductance. 


Eliminating  i6  between  eqs.  (2)  and  (6)  we  have 

(Rz  —  r)  ii  =  Ri±  —  rib 


(7) 


Again  eliminating  i6  between  eqs.  (2)  and  (4)  and  substituting 
the  value  of  i$  in  eq.  (7) 


(R,  -  r)t, 


(8) 


176  ELECTRICITY  AND  MAGNETISM 

Substituting  the  values  of  ii  and  t'4  in  terms  of  is  obtained  from 
eqs.  (3)  and  (5),  namely, 

r0  .     ,   L  di3 
*4  =  Rl3  +  R-dt 

/R  +  r<,\.     .  L  din 

"  =  "•  +  •*  -  (-R-)"  +  RTt 

in  eq.  8  there  results 

•       '  f 


Collecting  terms  we  have, 

R  +  r0          I  p(fi.  -  r)L  p  „  R 


—  r 


p. 

. 


Clearing  of  fractions, 

[(fi,  -  r)(fl  +  r0)  -  fir0)]i8  +  [(R*  -  r  -  R)L 

+  rfi,(/2  +r0)C  -  ^rroC]^  +  (fi,  -  B)rLC^f  =  0     (11) 

Assuming  now  that  t'3  is  an  alternating  current  of  the  form 

is  =  I  sin  ut 
and  substituting  this  in  eq.  (11)  there  results 

[(#2  -  r)(R  +  r0)  -  ^r0]  sin  ut  +  [(fi8  -  r  -  R}  L  +rR6 
(R  +  r0)  C  -  #rr0C]  7«  cosco^  -  (R3  -  R)  rLCIu*  sin  ut  =  0  (12) 

In  an  ordinary  measurement  in  which  C  is  expressed  in  micro- 
farads and  L  in  millihenries  the  last  term  is  of  the  order  10~9 
and  may  be  neglected  without  appreciable  error.     Since  eq.  (12) 
holds  for  all  values  of  t,  we  .have 
when  t  =  0, 

(#3  -  r  -  R)  L  +  [rRa(R  +  r0)  -  /frr0]   C  =  0 

whence 

[Rrr0  -  (R 


R3  -  R  -  r 
when 


~  r)(R  +  r0)  -  Rr0  =  Q  (14) 


ALTERNATING  CURRENT  BRIDGES  177 

which  is  seen  to  be  the  condition  for  a  steady  state  balance. 
If  /?3  =  R,  the  last  term  of  eq.  (12)  vanishes,  and  the  expression 
given  by  eq.  (13)  is  exact  and  reduces  to 


L  =  =  RR,C  =  R*C  (15) 

The  bridge,  when  used  in  this  manner,  is  well  adapted  to  the 
standardization  of  a  variable  inductance  such  as  Brooks  inductom- 
eter  but  is  not  well  suited  to  cases  in  which  the  values  of  L 
and  C  are  fixed  or  variable  by  steps,  since  it  is  impossible  to  adjust 
for  the  variable  state  balance  without  upsetting  that  for  steady 
states.  The  author  has  pointed  out  that  this  difficulty  may  be 
avoided  by  use  of  the  resistance  #4  as  shown  in  the  figure.  If 
identical  boxes  are  employed  for  r  and  R±  and  a  steady  state 
balance  obtained  with  R*  set  at  a  suitable  value,  then  the  variable 
state  balance  may  be  obtained  by  shifting  plugs  from  one  box  to 
the  other,  keeping  r  +  R*  constant.  If  an  equal  arm  bridge  is 
used,  this  has  the  effect  merely  of  adding  to  both  the  upper  and 
lower  right  hand  arms  of  the  bridge  the  value  of  7?4.  Eq.  (13) 
then  becomes 

j       [rr0  (R  -  R3  +  flQ  -  Rr(R*  -  R<)]C  _ 

Rs  _  fl4  _  R  _  r 

132.  Experiment  22.  —  Trowbridge's  Method  for  Self  Inductance. 
Connect  the  apparatus  as  shown  in  Fig.  100  using  the  telephone 
and  suitable  Oscillator  for  detector  and  energy  source  respectively. 
As  an  unknown  use  a  smoothly  variable  inductance,  and  set  the 
four  resistances  Ri,  R2,  Rs,  and  R  at  suitable  values,  e.g.,  500  ohms 
each.     C  should  be  a  subdivided  standard  condenser.     Measure 
the  unknown  for  several  settings  and  plot  its  calibration  curve. 
Replace  the  variable  inductance  by  one  of  fixed  value,  and  meas- 
ure it,  making  use  of  the  resistance  R^  as  explained  above. 

133.  Heydweiller's1    Network    for   Mutual  Inductance.  —  In 
Exp.  18,  a  method  due  to  Carey-Foster,  was  used  for  the  measure- 
ment of  mutual  inductnce  in  terms  of  capacitance.     The  essential 
feature  of  this  method  consists  in  balancing  the  charge  of  a  con- 
denser against  the  quantity  of  electricity  induced  in  the  secondary 
of  a  mutual  inductance  when  a  certain  current  change  takes  place 
in  the  primary.     This  balance  was  effected  by  discharging  the 
two  quantities  involved  in  opposite  directions  through  a  long 
period  ballistic  galvanometer.     While  this  circuit  is  satisfactory 

1  Annalen  der  Physik.,  vol.  53,  1894,  p.  499. 

12 


178 


ELECTRICITY  AND  MAGNETISM 


when  used  with  the  make  and  break  method  of  excitation,  it 
can  not  be  used  with  alternating  currents  since  there  is  no  way  of 
adjusting  the  time  constant  of  the  condenser  circuit. 

This  defect  was  overcome  by  Heydweiller  by  the  introduction 
of  the  resistance  S  as  shown  in  Fig.  101  and  a  very  satisfactory 
method  for  the  measurement  of  mutual  inductance  was  thus 
obtained.  The  resistance  P  includes  that  of  the  secondary 
coil  whose  self  inductance  is  L.  The  conditions  which  must  hold 


WW 

it 


VVNM/W 

it 


FIG.   101. — Heydweiller's  network  for  mutual  inductance. 

for  zero  current  through  the  galvanometer  may  be  obtained 
as  follows.  Designating  the  instantaneous  currents  through 
the  various  resistances  as  indicated  in  the  figure,  we  have 

1 4    ^    1%    ~T"   X\ 


Ri>  -  If* 


+ 


From  eqs.  (2)  and  (4)  we  have 

Ldii+Pi,- 

From  eqs.  (3)  and  (1) 


•S 


(2) 
(3) 

(4) 
(5) 
(6) 


Differentiating  eq.  (6)  and  substituting  in  eq.  (5),  there  results, 
on  collecting  terms, 


ALTERNATING  CURRENT  BRIDGES  179 

Since  an  alternating  E.M.F.  is  applied  to  this  circuit  the  current 
is  is  also  alternating  and  may  be  represented  by 

i3  =  I  sin  at ;  whence  ^,3  =  la  cos  at  (8) 

Substituting  these  values  in  eq.  (7)  we  have 

/D     I     Cf\-i  r-  KJf  ~\ 

—  M  - — = — -    la  cos  cofcH-    P  —  -^  \  I  sin  at  =  0     (9) 

Since  eq.  (9)  holds  for  all  values  of  t,  we  have 
when 

at  =  0,  L  -  M  (R  +  ^  =  0  or  M  =  Lp  f  Q  (10) 

/i  K  -\-  o 


when 


=,  P  -         =  0          or 
J  JKL 


It  is  thus  seen  that  there  are  two  conditions  which  must  be 
satisfied  in  order  that  there  should  be  no  deflection  of  the  galva- 
nometer, when  an  alternating  E.M.F.  is  applied.  The  second  of 
these  is  the  same  as  for  the  original  Carey-Foster  circuit,  which  is 
obtained  by  putting  S  =  O.  The  impossibility  of  satisfying  the 
first  condition  under  these  circumstances  is  obvious  for  then 
M  =  L,  an  inflexible  condition,  difficult  to  satisfy.  This  circuit, 
while  not  strictly  a  bridge,  resembles  one  in  that  two  balances 
conditions  are  necessary. 

134.  Experiment  23.  Heydweilkr's  Method  for  Mutual  Induc- 
tance.— Connect  the  apparatus  as  shown  in  Fig.  101.  For 
M  use  a  pair  of  coils  whose  relative  positions  may  be  varied,  and 
for  C,  a  subdivided  standard  condenser.  The  purpose  of  the 
experiment  is  to  determine  M  as  a  function  of  the  setting  of  the 
movable  coil.  As  source  and  detector  use  either  the  wire  inter- 
rupter and  vibration  galvanometer,  or  an  alternator  and  tele- 
phone. From  the  known  E.M.F.  of  the  source,  compute  the 
minimum  value  of  R  in  order  that  the  power  consumption  in  it 
should  not  exceed  4  watts  per  coil. 

Report. — Plot  M  as  a  function  of  the  scale  readings  of  the 
instrument.  In  computing  M  use  the  second  of  eq.  (10).  Check 
the  accuracy  of  your  results  by  substituting  in  the  first  of  these 
equations  and  note  the  constancy  of  the  values  for  L.  In  con- 
necting up  the  circuit  is  there  any  choice  as  to  which  coil  is 
used  as  the  primary?  Does  the  mutual  inductance  of  two  coils 
depend  upon  which  is  the  primary?  Explain. 


180 


ELECTRICITY  AND  MAGNETISM 


135.  Mutual  Inductance  by  Heaviside's  Bridge.1 — If  one  of 

the  arms  of  a  Wheatstone  bridge  is  inductive  while  the  other 
three  are  non-inductive  it  is  impossible  to  obtain  a  balance  since 
the  E.M.F.  across  the  inductive  arm  will  have  a  component  90 
deg.  out  of  phase  with  the  current  through  it,  and  the  E.M.F. 's 
at  the  galvanometer  corners  of  the  bridge  can  never  be  in  phase. 
It  was  pointed  out  by  Hughes  that  if  in  series  with  the  galva- 
nometer there  is  connected  the  secondary  of  a  variable  mutual 


FIG.  102. — Heaviside's  bridge  for  mutual  inductance. 


inductance,  the  primary  of  which  is  included  in  the  supply 
circuit,  an  E.M.F.  in  quadrature  with  this  current  and  hence 
opposite  in  phase  to  the  E.M.F.  due  to  the  self  inductance  of  the 
bridge  coil  may  be  obtained  and  a  balance  thus  secured. 

In  discussing  this  circuit,  Heaviside  pointed  out  that  a  more 
satisfactory  arrangement  results  if  the  secondary  of  the  mutual 
inductance  is  introduced,  not  in  the  galvanometer  circuit,  but  in 
the  arm  of  the  bridge  adjacent  to  that  containing  the  inductance 
under  consideration.  The  E.M.F.  thus  induced  in  LI  by  mutual 
inductance  may  be  made  to  compensate  the  difference  in  the 
E.M.F's  of  self  inductance  in  LI  and  L2.  Such  an  arrangement  is 
shown  in  Fig.  102.  The  balance  condition  is  obtained  as  follows: 

1  Phil.  Mag.,  vol.  19,  1910,  p.  497. 
The  Electrician,  vol.  76,  1885-86,  p.  489. 


ALTERNATING  CURRENT  BRIDGES  181 

Designating  by  i,  ii,  and  iz  the  instantaneous  supply  and  bridge 
currents  respectively,  the  following  equations  result. 

i  =  ii  +  ii  (1) 

D     *  Z>     '  /Q\ 

/t3ti    —    .ft4t2  V'-'/ 

& 

Eliminating  t  between  eqs.  (1)  and  (2),  we  have 
fliii  +  L!  ~  +  M  (^  +  - 


Substituting  in  eq.  (4)  the  value  of  iz  from  eq.  (3) 

D-         Tr          ij-/i    i    #3\~|<&i       ^2^3.         r    fi8dii      ,., 
R*i  +  [L,  +  M  (1  +  g-J  J-^  =     ^-  fc  +  L2  ^  -^      (5) 

Imposing  now  the  condition  for  steady  state  balance,  the  terms  in 
ii  vanish,  whence 


or 

M(R3  +  fa)  =  LZRS  -  L^4  (7) 

whence 


M 


If  an  equal  arm  bridge  is  used,  i.e.,  #3  = 

IT-      £,•-{*  (9) 


Campbell  has  suggested  a  simple  modification  of  this  bridge 
whereby  self  inductances  may  easily  be  measured  in  terms  of 
mutual,  provided  a  continuously  variable  standard  of  the  latter 
is  available.  This  inductance  is  introduced  at  I,  shown  short 
circuited  by  a  link  in  the  figure.  A  balance  is  first  obtained  with 
the  link  inserted.  Let  MI  be  the  reading  of  the  variable  standard 
for  this  setting.  Introduce  the  unknown  by  removing  the  link 
and  balance  again  varying  Ri  or  R%  to  compensate  for  the  added 
resistance  of  the  unknown  coil.  Let  M  2  be  the  new  reading  of 
the  standard.  Then,  for  an  equal  arm  bridge, 

M,  =  y2(L2  -  LO 
M2  =  %(L*  -  L!  +  Lx) 
whence 

Lx  =  2(M2  -  M,)  (10) 

where  Lx  is  the  unknown  self  inductance  to  be  measured. 


182  ELECTRICITY  AND  MAGNETISM 

A  further  simplification  results  if  L2  is  a  variable  inductance, 
for  then  the  first  balance  may  be  obtained  by  making  MI  zero  and 
adjusting  L2  until  it  is  equal  to  LI.  When  a  second  balance  has 
been  obtained,  Lx  is  simply  twice  the  value  of  M.  Neither  LI 
nor  L2  need  be  known.  This  method  is  particularly  useful  where 
a  number  of  inductances  of  the  same  order  of  magnitude  are  to 
be  measured. 

136.  Experiment  24.     Heaviside's  Bridge  for  Self  Inductance. — 
Connect  the  apparatus  as  shown  in  Fig.   102,  using  for  M  a 
variable    standard    of    mutual    inductance.     L2    should    be    a 
continuously    variable    self    inductance.     As    a    detector,    use 
phones,  vibration  or  A.C.  galvanometer  with  appropriate  source. 
Measure  a  series  of  self  inductances. 

Report. — Is  there  any  choice,  in  this  bridge,  as  to  which  of  the 
two  coils  of  the  mutual  inductance  is  used  as  the  primary? 
May  the  leads  to  the  primary  be  interchanged  at  liberty?  Could 
a  variable  state  balance  be  obtained  if  the  unknown  were  intro- 
duced in  the  arm  #4?  Explain. 

137.  Maxwell's  Bridge  for  Mutual  Inductance.1 — The  simplest, 
though  not  the  most  sensitive  bridge  for  the  measurement  of 
mutual  inductance  is  one  devised  by  Maxwell.     The  method 
consists  in  obtaining  the  mutual  inductance  of  a  pair  of  coils  in 
terms  of  the  self  inductance  of  one  of  them.     The  connections 
are  shown  in  Fig.    103.     In  the  discussion  of  the   Heaviside 
bridge,  Fig.  102,  it  was  pointed  out  that  a  balance  could  be 
obtained  by  introducing  in  the  coil  LI  by  mutual  inductance  an 
E.M.F.  which  would  compensate  for  the  difference  in  E.M.F.'s  in 
the  coils  LI  and  L2.     It  might  equally  well  have  been  said  that  the 
E.M.F.  in  the  coil  L2  balances  the  difference  between  the  E.M.F. 
in  LI  due  to  mutual  and  self  inductance.     If  the  relative  values 
of  the  currents  in  the  primary  and  secondary  of  M  are  changed 
these  E.M.F. 's  may  be  made  equal  without  the  use  of  the  coil  L2. 
This  is  the  method  employed  in  the  Maxwell  bridge,  and  is  accom- 
plished by  shunting  the  entire  bridge  by  the  resistance  R. 

Indicating  the  instantaneous  currents  as  shown  in  the  figure, 
the  equations  for  the  balance  condition  are  as  follows: 

i  =  ii  +  it  +  is  (1) 

BA-B*i-+'L&--MJ-  (2) 

Riii  =  #4*2  (3) 

Rit  =  (#3  +  Rfa  (4) 

1  MAXWELL,  Electricity  and  Magnetism,  vol.  2,  p.  365. 


ALTERNATING  CURRENT  BRIDGES 
Eliminating  i  between  eqs.  (1)  and  (2) 


di 


But,  from  eq.  (3) 
and,  from  eq.  (4) 


'  ~< 


R 


R          R, 


A/WVWWWVA 


FIG.   103. — Maxwell's  bridge  for  mutual  inductance. 

Substituting  these  values  in  eq.  (5)  there  results 


183 


(5) 


2       i        szi     /ox 

__a^__ 


RZ      i     RZ      |~   R* 

Imposing  the  condition  for  a  steady  state  balance,  the  terms  in  ii 
vanish,  and  we  have 

Since 
we  have 


~\- 


-f- 


184  ELECTRICITY  AND  MAGNETISM 

and  cq.  (7)  may  be  written 

L  (8) 


138.  Experiment   25.     Mutual  Inductance  in   Terms  of  Self 
Inductance   by   Maxwell's   Bridge.  —  Connect   the   apparatus   as 
shown  in  Fig.  103  using  for  M  several  pairs  of  coils  with  fixed 
mutual  inductances.     Operate  the  bridge  either  with  phones, 
vibration  or  A.C.  galvanometer,  and  appropriate  source  of  supply. 
After  the  determination  of  M  for  each  pair  of  coils,  interchange 
primary  and  secondary  and  check  your  result. 

Report.  —  Is  the  balance  as  sharply  defined  as  in  some  of  the 
bridges  previously  used?  Explain.  At  the  balance  point,  are 
the  currents  in  R  and  R  i  in  phase? 

139.  The  Mutual  Inductance  Bridge.  —  Figure  104  represents 
a  bridge  in  which  two  coefficients  of  mutual  inductance  may 
readily  be  compared  provided  one  of  them  is  a  variable  standard. 
Designating  the  various  parts  of  the  bridge  as  indicated  in  the 
figure,  the  balance  conditions  are  as  follows: 


D     •  ll/T  l  E>    •  71/f  2  /ON 

Rzii  -  Mi-^  =  R2iz  -  M2-j7  (2) 

Suppose  that  a  variable  state  balance  has  been  obtained  with 
the  secondaries  disconnected  and  the  galvanometer  joined 
directly  to  the  points  A  and  B.  This  balance  may  be  facilitated 
by  the  introduction  of  a  variable  inductance  in  series  with  either 
Ri  or  #2  as  the  case  may  require.  Under  these  circumstances 
eqs.  (1)  and  (2)  become  the  same  as  those  for  the  simple  induc- 
tance bridge,  namely 

R<i  +  L!  ^  =  R*i*  +  L2  ^j  (3) 

and 

R3ii  =  R*i*  (4) 

whence 

RS    •  /r\ 

*2  =  -rr  *i  (5) 

#4 

and 

diz  _  RS  dii 
dT  :=  #4  ~dt 


ALTERNATING  CURRENT  BRIDGES 

Substituting  in  eq.  (3)  we  have 

R^  +  L!  ^  =  ~^1  +  L2 13  ~^ 
Imposing  the  steady  state  balance,  we  have 


R 


185 
(6) 
(7) 


and 


Introduce  now  the  secondary  coils  as  shown  in  the  figure 
and  obtain  a  balance  by  adjusting  the  variable  standard.  This 
balance  indicates  that  the  E.M.F.'s  in  the  secondary  coils  are 
equal  and  opposite.  Since  no  current  flows  in  the  secondary 
coils,  the  currents  for  the  primaries  which  are  defined  by  eqs.  (3) 

and  (4)  are  unchanged.     Accordingly,  the  values  for  i2  and  -^ 

given  in  eq.  (5)  may  be  substituted  in  eq!  (1).     Subtracting  eq. 
(3)  from  eq.  (1)  then  gives 


This  bridge  is  distinguished  from  those  previously  studied  in  that 
three  balances  are  necessary.  This  may  seem  at  first  sight  to 
result  in  an  unduly  cumbersome  method,  but  experience  shows 
that  in  reality  it  is  a  relatively  simple  bridge  to  operate. 


FIG.  104. — Mutual  inductance  bridge. 

140.  Experiment  26.  Comparison  of  Two  Mutual  Inductances. 
Connect  the  apparatus  as  shown  in  Fig.  104,  using  a  pair  of 
phones  and  a  suitable  source  of  alternating  current.  Obtain  the 


186 


ELECTRICITY  AND  MAGNETISM 


three  balance  conditions  as  described  above,  for  several  different 

unknown  pairs  of  coils.     Check  your  results  by  interchanging 

primaries  and  secondaries  for  each  pair. 

Report. — Explain  why  it  is  permissible  to  introduce  an  extra 

inductance  in  series  with  RI  and  Rz.     Could  this  be  introduced 

in  R3  or  #4? 

141.  The  Frequency  Bridge.— In  all  of  the  bridges  thus  far 

discussed,  the  balance  condition  has,  in  no  case,  contained  a  term 

depending  upon  the  fre- 
quency. The  physical  sig- 
nificance of  this  is  that  these 
bridges  balance  independent 
of  the  frequency  and  hence 
the  form  of  the  impressed 
wave  is  of  little  consequence. 
A  bridge  will  now  be  studied 
which,  for  given  values  of  L 
and  Cj  can  be  balanced  for 
only  one  definite  frequency. 
The  connections  are  shown  in 

FIG.   105.— The  frequency  bridge.  Fig.    105.       Three  of  the  arms 

are    non-reactive,    while    the 

fourth  contains  an  inductance  and  a  condenser  in  series.  Let 
the  instantaneous  currents  through  the  upper  and  lower  arms  be 
ii  and  iz  respectively.  The  balance  conditions  are  then 

Riii  =  R*i*  (1) 


i  r 

-£  I  ii.<U  =  R& 


Eliminating  i'i,  we  have 


RtR, 


(2) 


(3) 


Imposing  the  steady  state  balance  condition,  which  may  be 
obtained  by  short  circuiting  C,  in  case  a  battery  and  ordinary 
galvanometer  are  used, 

=  0  (4) 


Assuming  that  a  pure  sine  wave  of  E.M.F.  is  applied  to  the  bridge, 
the  current  through  it  will  also  be  a  sine  wave  of  the  same  fre- 
quency as  the  source.  Let  the  current  i%  then  be  given  by 

iz  =  I  smut 


ALTERNATING  CURRENT  BRIDGES  187 

Then 

(*  T 

dlz  T  i      (    •     ^l  * 

— r-  =  7  co  cos  cot,  and    I  i^at  = cos  cot 

at  J  co 

Substituting  these  values  in  eq.  (4),  we  have 

L/co  cos  co£  -r  ^—  /  cos  co£  =  0  (5) 

C/co 

whence 

1 


•          •  a- vw 

Since  co  =  27rn,  where  ft  is  the  frequency, 

1 
n  =  9.^/Tr  (6) 


When  two  of  these  quantities  are  known,  the  third  may  be 
computed.  One  of  the  most  useful  applications  of  this  bridge  is 
the  measurement  of  frequency  using  a  subdivided  condenser  and 
a  continuously  variable  inductance.  In  case  a  complex  wave  is 
applied  to  the  bridge,  complete  silence  in  the  phones  can  not  be 
obtained  for  any  value  of  the  LC  product.  It  will  be  observed 
however,  that  the  relative  intensities  of  the  fundamental  and 
overtones  will  be  changed  as  L  is  varied  and  that  for  a  certain 
setting,  the  fundamental  will  disappear  while  the  overtones 
remain. 

In  Art.  110,  it  was  pointed  that  a  circuit  connected  for  parallel 
resonance,  possesses  a  very  large  impedance  for  the  particular 
frequency  to  which  it  resonances.  If  now  such  a  circuit  is  placed 
in  series  with  the  source,  and  adjusted  to  resonate  to  the  fre- 
quency of  the  fundamental  as  above  determined,  this  frequency 
may  be  suppressed  and  that  of  the  strongest  overtone  measured. 
Introducing  now  another  resonance  circuit  in  series  with  the 
source  to  suppress  this  overtone,  the  next  stronger  one  may  be 
measured  and  so  on.  In  this  way  a  qualitative  analysis  of  the 
wave  form  of  the  source  may  be  made. 

142.  Experiment  27.  Bridge  Method  for  Measuring  Frequency. 
Connect  the  apparatus  as  shown  in  Fig.  105,  using  for  C  a 
subdivided  mica  condenser,  and  for  L,  a  variable  standard  of 
inductance.  Determine  the  frequency  of  several  sources  of 
alternating  current,  using  the  phones  as  a  detector.  Determine 
first  the  fundamental  and  then  place  a  filter  circuit  in  series  with 
the  source  to  suppress  this  frequency  and  measure  the  frequency 
for  the  strongest  harmonic.  In  computing  the  frequency  by  eq. 


188 


ELECTRICITY  AND  MAGNETISM 


(6),  the  inductance  and  capacitance  must  be  expressed  in  henries 
and  farads  respectively. 

Report. — Compute  a  constant  for  the  right-hand  side  of  eq.  (6) 
which,  when  divided  by  \/LC  will  give  the  frequency  when  L 
is  expressed  in  millihenries  and  C  in  microfarads.  Compute  the 
inductance,  which,  when  used  with  a  capacitance  of  1  microfarad 
will  balance  the  bridge  for  a  frequency  of  60  cycles. 

143.  Circuits  of  Variable  Impedance. — In  the  bridges  studied 
thus  far  for  the  measurement  of  self  and  mutual  inductance,  the 
assumption  has  tacitly  been  made  that  the  only  E.M.F.'s  induced 
in  the  coils  are  those  due  to  the  primary  current.  For  example, 

in  the  case  of  mutual  inductance,  a 
varying  current  in  the  primary  pro- 
duces an  E.M.F.  in  the  secondary 
proportional  to  the  rate  of  change 
of  the  primary  current  hence  in 
quadrature  with  the  primary  cur- 
rent for  the  case  of  a  sine  wave. 
For  self  inductance,  the  coil  is  its 
own  secondary,  and  the  same  con- 
siderations hold  as  for  two  coils. 
The  direction  of  the  induced  E.M.F. 
is  counter  to  the  driving  E.M.F. 
while  the  current  is  rising,  and  in 
the  same  direction  when  it  is  fall- 
ing. The  power  associated  with  the 
induced  E.M.F.  at  any  instant  is  equal  to  the  product  of  this 
E.M.F.  and  the  current.  Energy  accordingly  is  alternately  stored 
in  the  electromagnetic  field  of  the  coil  and  returned  to  the  circuit. 
The  theory  shows,  in  fact,  that  this  occurs  at  a  frequency  twice 
that  of  the  driving  E.M.F. 

When  the  circuit  is  of  such  a  nature  that  energy  is  consumed 
by  the  coil  or  parts  connected  with  it  in  some  other  manner  than 
by  heat  developed  within  the  primary  coil,  the  quadrature  rela- 
tionship is  destroyed  and  the  impedance  of  the  coil  is  no  longer 
constant.  Among  the  more  important  causes  of  such  extraneous 
energy  consumption  are  hysteresis,  eddy  currents,  and  motion  of 
parts.  The  telephone  receiver  is  an  illustration  of  such  a  circuit. 
For  simplicity,  consider  a  coil  of  wire  C  wound  upon  a  bar  magnet 
M  near  one  end  of  which  is  placed  a  flexible  iron  diaphragm  D  as 
shown  in  Fig.  106.  When  an  alternating  current  flows  through 


FIG.  106.- 


D 

-Simplified  telephone 
receiver. 


ALTERNATING  CURRENT  BRIDGES  189 

C,  the  residual  magnetism  is  alternately  increased  and  decreased 
by  the  current  and  the  diaphragm  vibrates  with  the  same  fre- 
quency as  the  source.  In  addition  to  the  Joule  heat,  represented 
by  I2R,  developed  in  the  coil,  energy  consumptions  result  from 
the  three  causes  enumerated  above  in  the  following  manner. 

1.  Hysteresis. — The  E.M.P.  induced  in  the  coil  is  proportional 
to  the  rate  of  change  oj  the  magnetic  flux  through  it.     This  flux 
is  produced  by  the  magnetomotive  force  of  the  coil,  the  latter 
being  in  phase  with  the  current  and  proportional  to  it.     Because 
of  the  hysteretic  lag  of  flux  behind  the  magnetomotive  force,  the 
E.M.F.  is  no  longer  in  quadrature  with  the  current  but  has  a 
component  counter  to  the  current  which  results  in  a  continuous 
energy  consumption.     The  greater  the  area  of  the  hysteresis 
loop,  the  greater  the  lag  of  flux  behind  the  current  and  hence  the 
larger    the    energy    component   of   the    induced    E.M.F.     The 
hysteresis  loss  is  proportional  to  the  frequency. 

2.  Eddy  Currents. — The  magnet  M  may  be  regarded  as  a 
secondary  coil  consisting  of  a  single  turn  about  its  own  axis 
having  a  relatively  large  cross  section  and  a  low  resistance. 
The  changing  flux  through  this  turn  induces  in  it  an  E.M.F.  in 
quadrature  with  the  flux  and  the  resulting  current  is  known  as  a 
Foucault  or  eddy  current.     Because  of  the  small  self  inductance 
of  this  single  turn,  the  eddy  current  is  practically  in  phase  with 
the  induced  E.M.F.  producing  it.     The  eddy  current  may  in 
turn  be  considered  as  a  primary  which  induces  in  the  coil  a 
quadrature  E.M.F.     Except  for  the  hysteresis  lag,   this  final 
E.M.F.  in  C  is  counter  to  the  current  because  of  the  double 
quadrature   relationship,   and   hence   introduces  a  large  energy 
consumption.     Viewed    from    the    standpoint    of    Joule   heat 
developed    in    the    core    by    the    eddy    current,    this    loss    is 
proportional  to  the  square  of  the  frequency,  for  the  induced 
E.M.F.  producing  the  eddy  current  is  proportional  to  the  fre- 
quency and  the  heating  effect  of  a  current  is  equal  to  the  square 
of  the  E.M.F.  divided  by  the  resistance. 

3.  Motion  of  the  Diaphragm. — The  effect  of  motion  of  the 
diaphragm  may  be  understood  by  the  following  considerations. 
Suppose  a  sound  wave  strikes  the  diaphragm.     The  varying  air 
pressures  cause  it  to  vibrate  and  in  so  doing,  the  air  gap  between 
it  and  the  magnet  is  changed  and  hence  the  reluctance  of  the 
magnetic  circuit  of  the  magnet.     This  introduces  a  change  in 
flux  through  the  coil  which  induces  an  E.M.F.  within  it.     In 


190  ELECTRICITY  AND  MAGNETISM 

fact  this  is  the  principle  of  the  "  magneto-phone "  which  is  often 
employed  where  accurate  reproduction  is  more  essential  than 
energy  delivered.  As  regards  the  magnitude  of  the  E.M.F. 
induced  in  the  coil  and  the  phase  relation  between  it  and  the 
motion  of  the  diaphragm,  it  makes  no  difference  whether  the 
motion  is  produced  by  a  sound  wave  or  by  a  current  through  C. 
In  the  latter  case,  the  energy  of  the  wave  must  be  supplied  by 
the  current,  and  the  law  of  conservation  of  energy  requires  that 
the  induced  E.M.F.  due  to  the  motion  of  the  diaphragm  must 
have  a  component  counter  to  the  current  to  account  for  this 
consumption. 

If  an  alternating  current  of  intensity  I  is  passed  through  the 
coil,  and  the  power  delivered  to  the  coil  is  measured  by  appropri- 
ate means,  it  is  found  that  this  is  much  larger  than  would  be 
computed  from  PR  when  R  is  determined  by  using  direct  cur- 
rent. On  the  other  hand  we  may  define  a  resistance  Re  such  that 

Watts  =  I2Re. 

Re  is  called  the  "effective"  resistance  of  the  coil.  It  is  the  resis- 
tance of  a  fictitious  coil,  free  from  hysteresis,  eddy-current  and 
motional  reactions,  which  consumes  the  same  power  with  a  given 
current.  Again  the  effective  resistance  may  be  written 

Re  =  R  +  RH  +  RE  -\-  RM 

where  the  last  three  terms  represent  the  parts  contributed  by 
hysteresis,  eddy  currents  and  diaphragm  motion  respectively, 
and,  it  is  customary  to  speak  of  the  resistance  due  to  hysteresis, 
eddy  currents,  etc.  In  a  similar  manner,  the  E.M.F.'s  induced 
in  the  coil  by  hysteresis,  eddy  currents  and  motion  will  have 
components  in  quadrature  with  the  current,  and  will  change  the 
apparent  inductance  of  the  coil,  and  it  is  customary,  in  an 
analogous  manner,  to  speak  of  the  inductance  due  to  hysteresis, 
eddy  currents,  etc. 

Kennelly  and  Pierce1  have  made  a  detailed  study  of  the 
motional  characteristics  of  telephone  receivers  and  have  shown 
how  their  performance  in  practice  may  be  predetermined  from 
simple  measurements.  The  receiver  to  be  studied  was  placed 
in  one  of  the  arms  of  an  inductance  bridge  and  its  effective  resis- 
tance and  inductance  measured  for  a  wide  range  of  frequencies, 
first  with  the  diaphragm  clamped,  and  again  when  free  to  move. 

1  Proc.  Am.  Acad.  of  Sci.,  vol.  48,  p.  131,  1912. 


ALTERNATING  CURRENT  BRIDGES  191 

The  difference  between  the  corresponding  values  for  the  same 
frequency  were  called  " motional  resistance"  and  " motional 
inductance"  respectively.  The  latter  when  multiplied  by  the 
frequency  for  which  they  were  determined,  gave  the  "  motional 
reactance  "  for  that  frequency.  Interesting  results  were  obtained 
for  frequencies  near  that  corresponding  to  the  natural  period  of 
the  diaphragm.  For  example,  4he  curve  showing  the  motional 
resistance  as  a  function  of  the  frequency  closely  resembles,  near 
the  resonance  frequency,  the  curve  in  optics,  showing  the  varia- 
tion of  the  index  of  refraction  with  frequency  in  the  neighborhood 
of  an  absorption  band,  while  that  for  motional  reactance  exhibits 
a  sharp  minimum  at  this  point. 

144.  Experiment    28.     Motional    Impedance    of   a    Telephone 
Receiver. — Connect  the  apparatus  as  shown  in  Fig.  64  substi- 
tuting for  Lx  the  receiver  to  be  studied.     Use  an  equal  arm 
bridge  making  Ri  an  R2  approximately  equal  to  the  direct  current 
resistance  of  the  receiver.     Energize  the  bridge  with  a  Vreeland 
oscillator  which  has  previously  been  calibrated  for  frequency, 
and   use  a  pair  of  head  phones  as  a  detector.     Place  an  elec- 
trostatic voltmeter  across  the  output  coil  of  the  oscillator  and 
maintain   a   constant   voltage   on   the   bridge    throughout   the 
experiment.     Determine  roughly  the  natural  period  of  the  dia- 
phragm of  the  receiver  by  varying  the  frequency  of  the  oscillator 
keeping  the  voltage  approximately  constant  by  noting  at  what 
frequency  the  response  is  loudest.     Introduce  a  small  wedge  be- 
tween the  diaphragm  and  the  cap  to  prevent  motion  and  measure 
the  resistance  and  inductance  for  a  range  of  frequencies  above 
and  below  the  resonance  frequency.     Remove  the  wedge  and 
repeat  with  the  diaphragm  moving. 

Report. — Plot  curves  showing  the  variation  of  resistance  and 
reactance  with  frequency  for  both  blocked  and  moving  dia- 
phragm. Subtract  the  former  from  the  latter  and  thus  obtain 
the  "motional"  resistance  and  reactance  and  plot  each  as  a 
function  of  frequency. 

145.  Power  Factor  and  Capacity  of  Condensers.1 — In  a  perfect 
condenser,  that  is,  one  without  absorption  or  leakage,  the  phase 
of  the  current  is  90°  ahead  of  the  E.M.F.  impressed  across  its 
terminals.     Although  many  condensers  approximate  the  ideal,  it 
is  only  with  well  insulated  air  condensers  that  this  condition  may 

1  GROVER,  Bull.  U.  S.  Bureau  of  Standards,  vol.  3,  1907,  p.  371. 
WIEN,  Wiedemanris  Annalen,  vol.  44,  1891,  p.  689. 


192 


ELECTRICITY  AND  MAGNETISM 


be  regarded  as  actually  realized.  In  condensers  having  a  dielec- 
tric made  of  paper  impregnated  with  parafine  or  beeswax  there  is 
an  appreciable  component  of  the  current  in  phase  with  the 
E.M.F.  In  such  a  condenser  there  is  a  measurable  amount  of 
energy  absorption  which  appears  as  heat  in  the  dielectric,  and  as 
far  as  phase  relations  are  concerned,  it  may  be  regarded  as  a 
perfect  condenser  with  a  small  ficticious  resistance  in  series  with 
it.  In  Fig.  107a,  let  C  represent  the  equivalent  perfect  condenser, 
and  p  the  fictitious  series  resistance.  The  vector  diagram  107  6 
represents  the  phase  relations  for  such  a  circuit.  OE  is  the 


a  E 

FIG.  107. — Phase  diagram  for  a  condenser. 

impressed  E.M.F.  and  01  the  current  which  falls  short  of  the 
90°  lead  by  the  angle  6  which  is  designated  as  the  phase  differ- 
ence of  the  condenser.  0  is  the  phase  angle  as  ordinarily  defined 
and  the  power  factor  is  then 

P.F.  =  cos  0  =  sin  6. 


It  is  obvious  from  the  figure  that 


tan  6  =  pCco 


(1) 


A  simple  bridge  method  has  been  devised  by  Wein  by  which 
both  the  capacitance  and  the  power  factor  of  an  imperfect  con- 
denser may  be  simultaneously  measured  provided  there  is  avail- 
able for  comparison  purposes  another  condenser  which  shows  no 
absorption.  Such  a  bridge  is  illustrated  in  Fig.  108,  where  Ci  is 
the  perfect  condenser  and  €2  the  one  with  ficticious  resistance  p 
to  be  studied.  In  series  with  these  condensers  are  placed  the 
small  finely  adjustable  resistances  TI  and  r2.  The  purpose  of 
these  is  to  bring  about  equality  of  phase  in  the  currents  through 
the  upper  and  lower  branches  of  the  bridge.  It  is  clear  that, 


ALTERNATING  CURRENT  BRIDGES 


193 


without  them,  if  one  of  the  condensers  possesses  an  equivalent 
resistance  while  the  other  does  not,  the  potential  differences 
from  D  to  B  and  E  to  B  can  not  be  equal  and  in  phase  at  the  same 
time.  Accordingly  a  perfect  balance  of  the  bridge  can  not  be 
obtained.  If,  however,  a  suitable  resistance  ri  is  introduced  in 
series  with  C\  of  such  a  value*  that  the  time  constant  of  the  arm 
DB  equals  that  of  EQ,  this  difficulty  is  obviated.  In  practice 
it  is  generally  more  convenient  to  introduce  rz  also  and  take 
account  of  it  in  deducing  the  balance  conditions. 


FIG.  108. — Bridge  for  measuring  phase  difference  of  a  condenser. 

The  equations  for  balance  may  be  derived  in  the  following 
manner,  calling  ii  and  i2  the  instantaneous  currents  in  the  upper 
and  lower  arms  respectively. 

(2) 

W 

Eliminating  i%  we  have 


*1  =  —  I 


—     i2dt  +  (p  +  r2)i'2 


(3) 


If.,,         -        Ri  1    f- 
jrj  **'+**- fa J* 


(p  +  r2) 


Imposing  the  condition  for  a  steady  balance,  namely 

Ri          rl 


rz  +  P 


there  results 


(4) 


(5) 


(6) 


The  phase  difference  6  may  be  obtained  as  follows: 

13 


194  ELECTRICITY  'AND  MAGNETISM 

Combining  eqs.  (5)  and  (6)  we  have 


Multiplying  numerator  and  denominator  on  the  left  by  o>  and 
clearing  of  fractions,  there  results 

CWi  =  C2o>(r2  +  p)  (8) 

Referring  to  Fig.  1076  and  solving  eq.  (8) 

tan  6  =  C2cop  =  Cicori  —  C2cor2  (9) 

146.  Experiment  29.     Measurement  of  Phase  Difference  and 
Capacitance  of  a  Condenser.  —  Connect  the  apparatus  as  shown  in 
Fig.  108.     Ci  is  a  standard  mica  condenser  whose  phase  differ- 
ence is  regarded  as  negligible,  and  C2  is  a  telephone  condenser 
with  paraffine  paper  dielectric  whose  phase  difference  and  capaci- 
tance are  to  be  determined  as  a  function  of  the  frequency.     The 
resistances  TI  and  r2  are  small  in  value  and  should  be  joined  by  a 
slide  wire  for  fine  adjustment.     As  a  source  of  power  use  an 
oscillator  giving  a  pure  wave  form  whose  frequency  may  be 
varied  over  a  considerable  range,  such  as  the  Vreeland,  with 
phones  as  detector.     In  obtaining  a  balance,  set  r2  =  0  and  get 
as  good  silence  as  possible.     Introduce  such  a  value  of  ri  as  makes 
the  best  improvement,  then  change  Ri  or  R%  and  again  adjust  r\ 
and  so  on  until  complete  silence  is  reached.     Keep  r2  as  small  as 
possible.     Make  a  series  of  balances  using  as  wide  a  range  of 
frequencies  as  may  be  obtained  from  the  oscillator. 

Report.  —  Plot  capacity  and  phase  difference  of  the  unknown 
condenser  as  a  function  of  the  frequency.  Show  that  in  a 
perfect  condenser  the  current  leads  the  E.M.F.  by  90°.  Define 
Power  Factor. 

147.  Resistance   of  Electrolytes.  —  The  measurement  of  the 
resistance  of  an  electrolyte  offers  special  difficulties  not  encoun- 
tered   in    determining    the    resistance    of    metallic    conductors. 
This  is  due  to  the  fact  that  current  is  carried  through  a  solution 
by  virtue  of  the  migration  of  ions,  a  double  procession  in  opposite 
directions.     These  are  deposited  on  the  electrodes,  where  sec- 
ondary chemical  reactions  often  take  place.     In  general,   the 
deposits  on  the  electrodes  set  up  counter  E.M.F.  's  in  the  cell 
which  affect  the  measurements  in  the  same  manner  as  added 
resistance.     Obviously  then  an  electrolytic  resistance  can  not  be 
measured  by  a  Wheatstone  bridge  employing  direct  currents, 


ALTERNATING  CURRENT  BRIDGES 


195 


If  an  alternating  current  is  used  this  effect  is  eliminated  since 
the  counter  E.M.F.  is  with  the  bridge  current  during  one  half  of 
the  cycle  and  opposite  to  it  during  the  other. 

In  case  an  electrolyte  is  measured  in  which  a  gas  is  formed  at 
one  of  the  electrodes  a  further  complication  is  introduced  since 
the  cell  behaves  as  though  it  contains  capacitance.  This 
results  from  the  fact  that  a  gas  layer  separates  the  liquid  from  the 
electrode  thus  forming  a  condenser.  Since  the  gas  layer  is,  in 
general,  very  thin,  a  capacitance  of  considerable  magnitude  may 


FIG.  109.  —  Bridge  for  electrolytic  resistance. 

result.  The  cell  then  behaves  like  a  condenser  and  resistance 
in  parallel,  and  it  must  be  so  regarded  when  connected  in  one 
of  the  arms  of  a  bridge.  The  resistance  in  the  adjacent  bridge 
arm  must  also  be  shunted  by  a  condenser  else  a  balance  can  not 
be  obtained.  Such  a  bridge  is  shown  in  Fig.  109,  where  #4  and 
Cz  represent  the  resistance  and  capacitance  of  the  electrolytic  cell, 
and  #3  and  C\  its  counterpart  in  the  adjacent  arm.  Designating 
the  currents  as  indicated  in  the  figure,  we  have 

(1) 
(2) 
(3) 
(4) 


*i  =  *s  -h 

*2  =  *4  + 
#li'l  =  #2*2 
#3*3  =  #4*4 

It    =    #3*3 


77-  I  *6^    =    #4*4 


(5) 
(6) 


196 


ELECTRICITY  AND  MAGNETISM 


Eliminating  z'5  between  eqs.  (1)  and  (5)  and  z'6  between  eqs.  (2) 
and  (6)  there  results 


n  T?     3 
Ci/t3— r. 


=  14 


(7) 


(8) 


Substituting  in  eq.  (8)  the  values  of  i2  and  i4  from  eqs.  (3)  and  (4) 
and  eliminating  i\  between  the  resulting  equation  and  eq.  (7) 
we  have 


* 
*, 


,  , 


Imposing  now  the  condition  for  steady  state  balance,  i.e., 

R\  _  R$ 

R-2  R! 

we  have 


(10) 


= 
Cz        RI 


(11) 


In  carrying  out  measurements  of  the  resistance  of  solutions, 
the  design  of  the  electrolytic  cell  is  a  matter  of  considerable 


FIG.  110. — Cell  for  measurement  of  electrolytic  resistance. 

importance.  It  has  been  found  that  different  electrolytes  require 
different  types  of  cells  and  even  for  the  same  electrolyte  a  given 
cell  is  not  always  suited  to  wide  ranges  of  concentration.  For 
example,  polarization  may  occur  in  some  cases  unless  the  elec- 
trodes are  platinized,  and  in  other  cases  platinized  electrodes 


ALTERNATING  CURRENT  BRIDGES  197 

appear  to  act  calclytically  and  assist  chemical  action.  Again 
platinized  electrodes  may,  because  of  their  spongy  nature,  absorb 
so  much  of  the  electrolyte  as  to  cause  errors  in  measurement  when 
used  later  with  solutions  of  a  different  nature  or  concentration. 

Figure  110  shows  a  cell  designed  by  Dr.  Washburn  and  manu- 
factured by  the  Leeds  and  Ncvrthrup  Co.  The  electrodes  are  of 
platinum  and  are  mounted  by  sealing  their  supporting  wires  into 
tubular  glass  stems.  These  wires  project  through  the  seals  and 
connections  with  them  are  made  by  filling  the  stems  with  mercury. 
Side  tubes,  above  and  below  the  electrodes  respectively,  are 
attached  for  filling  and  washing  out  the  cell.  These  tubes  are 
bent  so  as  to  form  supports  for  holding  the  cell  in  a  suitable  bath 
for  maintaining  a  constant  temperature. 

148.  Experiment  30.  Resistance  of  Electrolytes. — Connect 
the  apparatus  as  shown  in  Fig.  109,  placing  the  solution  in  a  cell 
specially  designed  for  the  purpose.  Energize  the  bridge  with  the 
Vreeland  oscillator  and  detect  the  balance  with  a  telephone 
receiver.  Determine  the  resistance  of  a  series  of  solutions  fur- 
nished by  the  instructor.  Measure  the  dimensions  of  the  cell  and 
the  distance  between  electrodes  and  compute  the  specific  resistance 
of  each  solution. 

Report. — Explain  why  a  bridge  can  not  be  balanced  using  direct 
currents  when  it  contains  an  electrolytic  cell.  What  is  the 
essential  difference  between  metallic  and  electrolytic  conduction? 


CHAPTER  XIII 
CONDUCTION  OF  ELECTRICITY  THROUGH  GASES1 

149.  Electrons. — When  a  high  tension  discharge  passes  between 
electrodes  sealed  into  a  partially  evacuated  vessel,  the  gas 
becomes  luminous  showing  a  series  of  highly  colored  glows 
which  are  often  very  beautiful.  If  the  pressure  is  sufficiently 
reduced,  a  series  of  streams  appears,  proceeding  in  straight  lines 
from  the  cathode.  These  streams  are  known  as  "  cathode  rays," 
and  are  found  to  be  independent  of  the  position  of  the  anode,  and 
often  penetrate  regions  occupied  by  other  glows  in  the  tube. 

The  researches  of  modern  physics  have  shown  that  these  rays 
are  streams  of  discreet  particles  of  negative  electricity,  called 
"  electrons."  Their  properties  do  not  depend  upon  the  material 
of  the  electrodes  nor  the  nature  or  presence  of  the  gas  through 
which  the  discharge  takes  place.  They  may  be  produced  from  all 
chemical  substances,  and  consequently  must  play  an  important 
part  in  the  structure  of  matter.  The  velocities  with  which  they 
move  through  the  tube  vary  from  one-thirtieth  to  one-third  that 
of  light.  The  ratio  of  the  charge  of  an  electron  to  its  mass  is  con- 
stant and  is  equal  to  1.77  X  107  electromagnetic  units  per  gram. 
The  charge  of  an  electron  is  1.5  X  10~20  electromagnetic  units  and 

the  mass  is  about.,  ortn  that  of  the  hydrogen  atom.     The  radius 

l,oUU 

of  an  electron  is  estimated,  at  1.9  X  10~13  cms.,  which  is  about 
Kr.  nnn  that  of  the  atom.  For  many  years  the  mass  has  been 

OUjUUU 

regarded  as  purely  electromagnetic  in  character;  that  is,  while 
exhibiting  inertia,  it  shows  no  gravitational  attraction  in  the  sense 
possessed  by  ordinary  matter.  Recently,  however,  certain 
experimental  and  theoretical  evidence  has  been  produced  which 
makes  it  appear  likely  that  this  cannot  be  entirely  the  case. 

1  CROWTHER,  Ions,  Electrons  and  Ionizing  Radiations. 
McCLUNG,  Conduction  of  Electricity  through  Gases  and  Radioactivity. 
MILLIKIN,  The  Electron. 

THOMSON,  Discharge  of  Electricity  through  Gases. 
TOWNSEND,  Electricity  in  Gases. 

198 


CONDUCTION  THROUGH  GASES  199 

Many  attempts  have  been  made  to  discover  evidence  of 
quantities  of  electricity  smaller  or  larger  than  the  electron,  but 
none  smaller  have  ever  been  found.  In  fact,  when  quantities 
comparable  to  the  electron  have  been  isolated,  they  have  always 
proved  to  be  exact  integral  multiples  of  it.  The  evidence  points 
to  the  conclusion  that  electricity  is  atomic  in  structure  and  that 
the  smallest  possible  element  fs  the  electron,  which  thus  con- 
stitutes our  natural  unit  of  electricity.  Electric  currents  through 
conductors,  as  we  know  them  in  every  day  practice,  are  simply 
streams  of  electrons  through  or  between  the  atoms  and  molecules 
making  up  the  conducting  body. 

150.  Conductivity  of  Gases. — A  gas  in  its  normal  state  is  one 
of  the  best  insulators  known.  This  may  be  shown  by  mounting 
a  gold  leaf  electroscope  inside  an  inclosed  space,  and  allowing 
only  a  small  rod  carrying  a  polished  knob,  for  the  purpose  of 
charging,  to  project  out.  If  the  support  carrying  the  electro- 
scope is  well  insulated  from  the  container,  the  electroscope  will 
remain  charged  for  a  long  time,  showing  that  the  air  or  what- 
ever gas  surrounds  the  electroscope  is  a  poor  conductor  of 
electricity. 

If,  however,  X-rays  are  allowed  to  shine  through  the  enclosure, 
or  if  a  small  quantity  of  some  radio-active  substance  such  as 
thorium  or  radium  is  placed  inside  it,  or  again  if  the  products  of 
combustion  of  a  flame  are  drawn  through  it,  it  is  then  found  that 
the  gold  leaves  collapse  quite  rapidly,  indicating  that  the  gas  has 
lost  its  insulating  properties.  That  the  leakage  has  taken  place 
through  the  air  and  not  across  the  insulating  support  may  be 
shown  by  using  a  second  chamber  connected  with  the  electrom- 
eter enclosure  by  a  glass  tube,  and  introducing  the  X-rays,  the 
radio  active  substance  or  other  agent  into  this,  and  then  drawing 
the  air  thus  acted  upon  into  the  first  chamber.  The  same  effects 
are  observed.  However,  if  glass  wool  is  introduced  in  the  con- 
necting tube,  or  if  the  air  is  passed  between  two  insulated  plates 
connected  to  a  battery  before  entering  the  electrometer  chamber, 
it  is  found  that  its  insulating  properties  are  restored.  Experi- 
ments of  this  sort  as  well  as  many  others  of  an  entirely  different 
nature  have  shown  that  the  conduction  of  electricity  through 
gases  is  due  to  carriers  of  electricity,  and  that  the  carriers  are  of 
two  distinct  types,  positive  and  negative;  the  former  are  similar 
to  the  carriers  of  electricity  through  solutions  and  are  called 
positive  ions,  while  the  latter  are  either  negative  ions  or  electrons. 


200  ELECTRICITY  AND  MAGNETISM 

151.  Structure  of  the  Atom. — To  explain  the  phenomena  of 
the  conductivity  of  gases,  it  is  necessary  first  to  make  a  brief 
statement  concerning  the  structure  of  the  atom.     While  our 
knowledge  is  far  from  complete,  it  is  well  established  that  the 
atom  consists  of  a  nucleus  of  positive  electricity,  about  which 
revolve  in  closed  orbits,  electrons,  in  much  the  same  way  that 
the  planets  revolve  about  the  sun,  and  that  the  relative  dimen- 
sions of  electrons,  nucleus  and  orbits  are  about  the  same  as  in 
the  solar  system.     The  number  of  electrons  present  in  a  given 
atom  has  been  estimated  in  various  ways,  and  while  the  results 
are  not  entirely  in  agreement,  it  is  probable  that  it  is  the  same 
as  the  atomic  number,  that  is,  its  number  in  the  list  of  elements 
arranged  in  order  of  ascending  atomic  weights.     The  atomic 
number,  except  for  the  case  of  hydrogen,  is  approximately  half 
its  atomic  weight.     Since  the  atom  as  a  whole  is  neutral,  it  is 
necessary  that  the  positive  nucleus  should  have  a  charge  equal 
to  ne,  where  e  is  the  charge  of  the  electron  and  n  the  number  of 
electrons.     The  shape  of  the  orbits,  the  law  of  force  between 
nucleus  and  electron,  and  even  the  conditions  of  stability  are 
problems  which  have  not  yet  been  solved,  but  are  now  being 
attacked  from  many  angles. 

When  external  agencies  such  as  X-rays,  ultra  violet  light,  radia- 
tions from  radio  active  materials,  etc.,  act  upon  a  gas,  it  is  found 
that  the  atomic  structure  is  broken  up.  One  or  more  electrons 
may  be  torn  away  from  the  system  leaving  it  with  an  excess  of 
positive  electricity.  We  thus  have  present  in  the  gas  positive 
ions  and  negative  electrons.  The  gas  is  then  said  to  be  ionized, 
and  the  means  by  which  this  condition  is  brought  about  is 
called  the  " ionizing  agent."  If  two  electrodes  are  introduced, 
and  a  difference  of  potential  is  maintained  between  them,  the 
electrons  move  to  the  positive  electrode,  and,  entering  it,  pass  on 
through  the  external  metallic  circuit.  The  positive  ions,  on  the 
other  hand,  move  to  the  negative  electrode  and  receive  electrons 
from  it,  thus  becoming  again  neutral  molecules.  Unless  an 
ionizing  agent  acts  continuously,  the  current  through  the  circuit 
will  persist  only  until  the  ions  and  electrons  have  been  removed 
from  the  gas. 

152.  The  lonization  Current. — Suppose  now  that  an  ionizing 
agent  is  acting  continuously  upon  a  gas  in  an  ionization  chamber, 
as  an  arrangement  such  as  that  just  described  is  called.     At  first 
it  might  be  supposed  that  if  the  agent  acts  long  enough  all  of 


CONDUCTION  THROUGH  GASES  201 

the  atoms  would  be  ionized.  This,  however,  is  not  the  case;  for, 
due  to  their  undirected  heat  motion,  ions  and  electrons  collide, 
and  recombine.  When  the  rate  of  recombination  is  equal  to 
that  of  ionization,  a  steady  state  is  reached  where  only  a  definite 
fraction,  usually  a  very  small  number,  of  the  total  number  of 
molecules  are  in  the  ionized  state.  If  the  difference  of  potential 
between  the  plates  is^varied,  &nd  the  current  between  them  is 
measured  and  plotted  as  a  function  of  voltage,  it  is  found  that 
the  current  increases  with  the  voltage  almost  linearly  at  first,  in 
accordance  with  Ohm's  law;  but  for  higher  voltages,  the  curve 
is  concave  downward  and  when  a  certain  voltage  has  been 
reached,  no  further  increase  in  current  can  be  obtained,  unless 
the  voltage  is  raised  to  very  large  values.  The  constancy  of  the 
current  is  due  to  the  fact  that  all  of  the  ions  and  electrons  pro- 
duced are  swept  out  by  the  field.  This  current  is  spoken  of  as 
the  " saturation  current,"  from  the  similarity  between  the  shape 
of  this  curve  and  the  magnetization  curve  for  iron.  The  voltage 
at  which  the  horizontal  part  of  the  curve  begins  is  called  the 
1 1  saturation  voltage . ' ' 

If  the  distance  between  the  electrodes  is  increased,  it  might,  by 
analogy  with  metallic  conductors,  be  thought  that  the  saturation 
current  would  be  reduced  because  of  the  increased  path  the  ions 
and  electrons  must  travel.  It  is  found,  however,  that  the  cur- 
rent is  actually  increased.  This  is  because  there  is  a  larger 
number  of  gas  molecules  subjected  to  the  action  of  the  ionizing 
agent,  and  hence  more  carriers  are  produced.  Again,  it  is  found 
that  if  the  pressure  of  the  gas  is  increased,  the  ionization  current 
is  increased.  Both  of  these  facts  show  that  the  saturation  cur- 
rent through  a  gas  is  proportional  to  the  mass  of  the  gas  between 
the  electrodes. 

153.  Ionization  by  Collision. — If  the  voltage  between  the 
plates  of  the  ionization  chamber  is  increased  to  sufficiently  large 
values,  the  saturation  current  does  not  remain  constant  indefi- 
nitely, for  fields  may  be  reached  at  which  the  current  again 
begins  to  rise,  slowly  at  first  and  then  very  rapidly,  finally  result- 
ing in  a  disruptive  spark  accompanied  by  the  passage  of  a  current 
of  considerable  magnitude.  The  field  required  for  this  increased 
current  depends  upon  the  distance  between  electrodes,  their  size 
and  shape,  and  the  nature  and  pressure  of  the  gas.  For  air  at 
atmospheric  pressure  and  spherical  electrodes  of  moderate  dimen- 
sions, e.g.,  1  cm.  diameter,  it  is  of  the  order  of  10,000  volts  per 


202  ELECTRICITY  AND  MAGNETISM 

centimeter.  It  diminishes,  however,  as  the  pressure  is  reduced, 
and  is  most  conveniently  studied  at  pressures  below  10  milli- 
meters of  mercury. 

This  increase  in  current  is  due  to  the  fact  that  ions  are  produced 
by  collisions  taking  place  between  neutral  molecules  and  ions  as 
well  as  electrons  already  existing  in  the  gas.  The  mechanism 
of  this  process  is  somewhat  obscure,  but  it  is  clear  that  a  definite 
amount  of  energy  is  required  to  disrupt  a  neutral  atom.  The 
kinetic  energy  of  motion  of  the  ions  and  electrons  depends  upon 
how  far  they  have  moved  under  the  accelerating  field  before 
being  stopped  in  the  same  way  that  the  energy  of  motion  of  a 
freely  falling  body  depends  upon  the  distance  through  which  it 
has  fallen  before  being  arrested.  Thus,  as  the  pressure  of  the 
gas  is  reduced,  the  average  length  of  free  travel  is  greater  and  the 
acquired  energy  available  for  ionizing  purposes  is  increased. 
The  conductivity  of  a  gas  therefore  increases  as  the  pressure  is 
reduced.  Since,  however,  the  conductivity  depends  upon 
carriers  which  come  originally  from  neutral  molecules,  the  con- 
ductivity can  not  increase  indefinitely  with  decrease  of  pressure, 
for  the  effect  of  the  decreased  available  supply  will  eventually  be 
felt.  An  optimum  pressure  therefore  exists  at  which  the 
increased  range  for  acceleration  is  just  balanced  by  the  decreased 
supply  of  molecules.  For  air,  this  pressure  is  of  the  order  of  a 
few  tenths  of  a  millimeter  of  mercury.  A  further  decrease  in  the 
pressure  results  in  a  rapid  increase  in  the  resistance  of  the  gas. 
If  a  perfect  vacuum  could  be  obtained,  the  free  space  between 
electrodes  would  be  a  perfect  insulator.  While  this  is,  of  course, 
impossible,  it  is,  nevertheless,  easy  with  modern  methods  of 
evacuation  to  obtain  pressures  so  low  that  no  appreciable  dis- 
charge can  be  detected  with  the  highest  fields  available  in  the 
laboratory. 

154.  Experiment  31.  Resistance  of  a  Discharge  Tube. — The 
apparatus  consists  essentially  of  a  discharge  tube,  as  shown  in 
Fig.  Ill,  about  fifteen  inches  in  length  through  the  ends  of 
which  are  sealed  wires  attached  to  electrodes  of  relatively  large 
area.  It  is  connected  to  a  high  vacuum  pump  by  means  of  which 
the  pressure  may  be  reduced  to  any  desired  value.  A  manometer 
and  McLeod  gauge  are  also  joined  to  the  tube  to  measure  the 
pressure. 

Connect  a  small  high  tension  transformer  across  the  tube  to 
supply  the  voltage  for  the  discharge.  Place  an  electrostatic 


CONDUCTION  THROUGH  GASES 


203 


voltmeter  across  the  tube  and  an  A.C.  milliameter  in  series  with 
it.  The  impressed  voltage  may  be  controlled  by  a  series  resis- 
tance in  the  primary  circuit.  Starting  at  one  atmosphere,  reduce 
the  pressure  until  a  current  of  10  or  15  milliamperes  is  obtained 
through  the  tube.  Measure  the  required  voltage.  Take  a 
series  of  readings  at  various  pressures  measuring  the  voltage 

& 

To  Pump  and  Gauge 


Mill 
Ammeter 


High  Tension 
Transformer 


Supply 
FIG.   111. — Resistance  of  discharge  tube. 


required  to  maintain  a  definite  predetermined  current.  Com- 
pute the  resistance  of  the  tube  by  Ohm's  law.  Repeat  the  experi- 
ment using  twice  this  current. 

Report. — Plot  a  curve  showing  the  resistance  of  the  tube  as 
a  function  of  pressure.  Why  must  the  current  be  held  constant  in 
this  experiment?  Explain  the  operation  of  the  McLeod  gauge. 

155.  Phenomena  of  the  Discharge  Tube. — If  electrodes  are 
mounted  at  the  ends  of  a  tube  such  as  shown  in  Fig.  112,  con- 


204  ELECTRICITY  AND  MAGNETISM 

taining  air  at  ordinary  pressures  and  a  sufficiently  high  voltage 
is  impressed  between  them,  the  phenomenon  first  observed  is  the 
ordinary  spark  similar  to  that  between  the  electrodes  of  a  static 
machine.  If,  however,  air  is  gradually  removed,  the  sparks 
become  less  violent,  and  fine  streamers  of  bluish  color  are 
observed.  As  the  pressure  is  further  reduced,  these  streamers 
broaden  out  and  fill  the  entire  tube,  and  a  pink  color  appears 
at  the  anode.  With  further  exhaustion,  the  pink  color  extends 
some  distance  from  the  anode  and  dark  spaces  appear  in  the 
region  of  the  cathode.  When  the  pressure  has  been  reduced  to 


123  4  5 

FIG.  112. — Luminous  regions  of  discharge  tube. 

about  half  a  millimeter  of  mercury,  the  discharge  assumes  a  very 
characteristic  appearance.  Closely  surrounding,  but  not  quite 
touching  the  cathode,  is  a  thin  layer  of  luminosity  known  as  the 
cathode  glow.  Next  to  this  is  a  region,  from  which  no  light  is 
observed,  called  the  Crooke's  dark  space,  and  beyond  this  is  a 
rather  broad  region  of  luminosity  known  as  the  negative  glow. 
Following  this  is  another  non-luminous  region,  called  the  Faraday 
dark  space.  Between  this  dark  space  and  the  positive  electrode 
is  a  region  called  the  positive  column,  which  may  be  seen  as  a 
continuous  band  of  light  or,  under  certain  conditions  of  current 
and  voltage,  as  a  series  of  light  and  dark  striae.  The  positive 
column  seems  to  be  definitely  associated  with  the  anode,  for  if 
the  tube  is  increased  in  length  or  bent  into  a  curve,  the  positive 
column  increases  or  bends  with  it,  while  the  other  parts  of  the 
discharge  remain  fixed  and  are  thus  shown  to  be  associated  with 
the  cathode.  These  luminous  regions  are  indicated  in  Fig.  112. 
If  the  pressure  is  still  further  reduced,  the  striae  of  the  positive 
column  become  fewer  in  number  and  wider  in  extent  and  finally 
disappear.  The  regions  associated  with  the  cathode  also  become 
larger  and,  with  the  disappearance  of  the  positive  column,  the 
dark  spaces  fill  nearly  the  entire  tube.  With  sufficient  ex- 
haustion, the  Crookes  dark  space  completely  fills  the  tube, 
and  the  voltage  required  for  a  passage  of  current  becomes 


CONDUCTION  THROUGH  GASES  205 

very  high.  At  this  stage,  the  walls  of  the  tube  fluoresce  bril- 
liantly with  colors  depending  upon  its  chemical  composition, 
being  bluish  for  soda}  and  bright  green  for  German  glass.  If 
the  exhaustion  is  carried  far  enough,  the  tube  becomes  a  non- 
conductor of  electricity.  . 

156.  Theory  of  the  Discharge.1 — Since  no  external  ionizing 
agent  is  acting,  it  is  obvious  that  the  discharge  is  maintained 
by  ions  produced  by  collision,  and  the  varied  distributions  of  the 
luminous  regions  indicate  that  the  electric  fields  and  the  velocities 
of  the  carriers  can  not  be  uniform  throughout  the  tube.  It  has 
not  yet  been  definitely  determined  whether  luminescence  arises 
from  ionization  of  neutral  molecules  or  whether  it  accompanies 
the  recombination  of  an  ion  and  an  electron  to  form  a  neutral 
molecule.  At  the  present  time,  the  evidence  seems  to  favor  the 
latter  hypothesis.  Another  widely  accepted  view  is  that  when  a 
molecule  has  been  shaken  up  by  collision  with  an  ion  or  electron 
to  such  an  extent  that  its  electronic  orbits  are  badly  distorted, 
but  not  disrupted,  light  emission  accompanies  its  return  to  the 
equilibrium  state.  On  the  latter  theory,  luminous  regions  do  not 
necessarily  coincide  with  regions  of  ionization.  Some  of  the  more 
important  phenomena  characterizing  the  several  regions  enu- 
merated above  are  the  following. 

1.  Cathode  Glow. — The  field  strength  in  this  region  is  large  and 
often  the  greater  part  of  the  entire  potential  difference  occurs  in 
this  limited  space.    The  magnitude  of  the  fall  in  potential  depends 
upon  the  nature  of  the  gas  and  the  material  of  the  electrode, 
ranging  from  470  volts  for  water  vapor  to  170  volts  for  argon  with 
platinum  electrodes.     If  metals  such  as  magnesium,  sodium,  or 
potassium  are  used,  much  smaller  values  are  obtained  because 
of  the  greater  ease  with  which  these  substances  emit  electrons. 
The  large  potential  gradient  here  is  caused  by  the  accumulation 
of  positive  ions  in  front  of  the  cathode.     Because  of  the  greater 
mobility  of  electrons,  they  rapidly  move  away  from  this  region 
thus  leaving  a  preponderance  of  positive  ions.     The  ionization 
is  caused  by  collision  of  the  positive  ions  either  with  gas  molecules 
or  the  cathode  itself. 

2.  Crookes  Dark  Space. — It  was  pointed  out  above  that  a 
certain  amount  of  energy  is  required  to  produce  ionization.     The 
electrons  from  the  cathode  glow  must  move  through  a  certain 

1  CROWTHEE,  Ions,  Electrons,  and  Ionizing  Radiations,  chap.  VI. 
TOWNSEND,  Electricity  in  Gases,  chap.  XI. 


206  ELECTRICITY  AND  MAGNETISM 

difference  of  potential  before  they  possess  the  requisite  kinetic 
energy  for  this  purpose.  The  Crookes  dark  space  represents  this 
distance  for  it  is  here  that  electrons,  liberated  in  the  cathode  glow, 
are  acquiring  the  necessary  energy  of  motion  to  produce  the 
ionization  of  the  negative  glow.  It  is,  in  general,  a  rough  measure 
of  the  mean  free  path  of  the  electrons.  No  ionization  occurs  in 
this  region  and  the  current  is  carried  almost  exclusively  by  the 
electrons. 

3.  Negative  Glow. — The  luminosity  of  this  region  is  due  to 
ionization  by  electrons  from  the  Crookes  dark  space.    The  positive 
ions  produced  here  move  slowly  out  of  the  negative  glow  into  the 
Crookes  dark  space  and  their  presence  reduces  the  potential 
gradient  to  such  an  extent  that  electrons,  originating  in  the  nega- 
tive glow,  do  not  gain  sufficient  speed  to  produce  ionization;  and 
hence,  after  those  entering  from  the  Crookes  dark  space  have  been 
stopped  by  the  ionization  process,  no  further  ionization  occurs. 

4.  Faraday  Dark  Space. — The   current  in  this  region  is  due 
largely   to   electrons   which   enter  it  from  the  negative  glow. 
Because  of  the  accumulation  of  electrons  in  the  negative  glow,  the 
potential  gradient  through  the  Faraday  dark  space  and  even  up  to 
the  anode  is  quite  large.     The  electrons  are  accordingly  accel- 
erated through  this  dark  space  and  when  they  have  attained 
velocities  sufficient  for  ionization,  the  positive  column  commences. 

5.  Positive   Column. — The    potential   gradient   is   practically 
constant  throughout  this  region  and  ionization  by  collision  may 
take  place  all  the  way,  resulting  in  a  uniform  column  of  light. 
Ordinarily,  however,  there  are  local  accumulations  of  positive 
ions,  which  result  in  a  decrease  in  the  potential  gradient  with  a 
consequent  reduction  in  the  acceleration  of  the  electrons.     There 
are  then  regions  in  which  the  velocities  are  too  small  to  produce 
ionization  and  the  striae   commonly  observed,  result.     Under 
these  circumstances,  the  positive  column  is,  to  a  certain  extent, 
a  repetition  of  the  phenomena  of  the  Crookes  dark  space,  and  the 
negative  glow. 

157.  Investigation  of  the  Field  Strength  at  Various  Points 
in  the  Discharge.1 — The  potential  at  any  point  in  a  tube  may  be 
determined  by  inserting  an  auxilliary  electrode.  A  small  plati- 
num wire  is  most  frequently  used  for  this  purpose.  If  the  region 
happens  to  be  one  of  high  potential,  the  wire  will  attract  to  it 
positive  ions  until  its  potential  is  the  same  as  that  of  its  surround- 

1  GRAHAM,  Wied.  Ann.,  vol.  64,  1898,  p.  49. 


CONDUCTION  THROUGH  GASES  207 

ings,  which  is  then  indicated  by  an  electrometer  to  which  the  wire 
is  attached.  Accurate  results  can  be  obtained  by  this  method 
only  when  there  is  a  plentiful  supply  of  ions  of  both  signs.  For 
example,  suppose  the  wire  is  introduced  near  the  anode,  where 
only  electrons  are  present.  The  forces  of  the  field  will  cause 
electrons  to  strike  the  wire  until  it  is  so  highly  charged  negatively 
that  no  more  can  react  it  beca&se  of  repulsion,  and  the  wire  thus 
has  a  negative  potential  considerably  in  excess  of  the  region  in 
which  it  is  placed.  If  positive  ions  also  were  present,  they  would 
be  drawn  to  the  wire  until  its  potential  is  the  same  as  the 
surrounding  region. 

If  two  test  electrodes  are  used,  the  field  strength  at  various 
points  through  the  discharge  may  be  determined  by  measuring 
the  potential  difference  between  them  and  dividing  by  their 
distance  apart.  Except  for  regions  close  to  the  electrodes,  where 
only  one  type  of  ion  is  present,  this  method  gives  reliable  results. 
Because  of  the  mechanical  difficulty  of  moving  a  pair  of  test  wires 
through  a  tube  with  fixed  electrodes,  it  is  more  convenient  to  use 
a  tube  with  fixed  test  wires  it  and  moveable  electrodes  as  shown  in 
Fig.  113.  The  anode  A  and  cathode  C  are  held  at  a  fixed  distance 
apart  by  means  of  a  glass  rod  d  with  flexible  leads  connecting  to 
the  seals  through  the  tube.  A  small  lug  of  iron  /  is  acted  upon  by 
a  magnet  so  that  the  electrodes  may  be  moved  along  the  tube, 
placing  the  test  wires  at  any  desired  part  of  the  discharge. 

158.  Experiment  32.  Measurement  of  Field  Strength  in  the 
Discharge  through  Air.  —  Connect  the  apparatus  as  shown  in  Fig. 


To  Electrometer 


To  Pump  and  Gauge 
FIG.   113. — Tube  for  measuring  potential  gradients. 

113,  using  as  a  source  of  power  either  a  battery  of  flash  light  cells 
or  a  motor  generator  set  giving  an  E.M.F.  of  about  1,000  volts. 
Include  a  graphite  resistance  in  series  with  the  tube  to  prevent 
arcing  when  the  conductivity  is  high.  Measure  the  difference 
of  potential  between  the  test  electrodes  by  means  of  an  electrom- 


208  ELECTRICITY  AND  MAGNETISM 

eter  which  has  been  checked  against  a  standard  voltmeter. 
Start  the  pump  and  note  the  character  of  the  discharge  from  the 
highest  pressure  at  which  a  current  can  be  maintained  to  the  best 
vacuum  that  the  pump  will  give.  An  E.M.F.  of  1,000  volts  is 
not  in  general  sufficient  to  start  the  discharge  although  it  will 
maintain  it,  once  it  is  going.  To  start  it  connect  a  small  spark  coil 
across  the  tube  with  an  air  gap  in  series  to  prevent  shorting  the 
generator  or  battery  through  the  secondary  of  the  coil.  Deter- 
mine the  field  strength  at  various  points  through  the  discharge 
for  two  pressures  (a)  the  highest  at  which  a  uniform  discharge 
can  be  maintained,  (6)  one  at  which  the  discharge  has  the 
characteristic  appearance  shown  in  Fig.  112.  Measure  the  pres- 
sures by  means  of  a  McLeod  gauge,  and  the  voltage  across  the 
tube  by  an  electrostatic  voltmeter. 

Report. — Indicate  by  sketches  the  character  of  the  discharge 
for  several  different  pressures.  Plot  field  strength  as  a  function 
of  distance  from  the  cathode  for  the  two  cases  studied.  Plot 
voltage  as  a  function  of  distance  from  cathode.  Obtain  the 
latter  from  the  area  under  the  field  strength — distance  curve. 

159.  Cathode  Rays. — It  was  pointed  out  above  that  when  the 
pressure  in  a  discharge  tube  has  been  reduced  to  a  certain  value, 
e.g.,  a  hundredth  of  a  millimeter  of  mercury,  the  character  of  the 
discharge  is  entirely  changed  from  that  represented  by  Fig.  112. 
The  positive  column  shrinks  back  and  disappears  entirely  and 
the  Crooke's  dark  space  occupies  the  entire  volume  of  the  tube. 
The  glass  now  shows  a  bright  fluorescence,  green  or  blue,  depend- 
ing upon  its  composition.  This  fluorescence  is  due  to  bombard- 
ment by  electrons  shot  out  from  the  cathode  or  the  region 
immediately  in  front  of  it.  They  travel  in  straight  lines  perpen- 
dicular to  the  cathode,  and  possess  many  interesting  proper- 
ties. For  example,  if  they  strike  a  piece  of  platinum  foil,  it  may 
be  heated  to  incandescence  by  their  bombardment,  or  if  they 
impinge  upon  substances  such  as  willimite,  calcium  tungstate, 
barium  platino-cyanide,  etc.,  they  cause  them  to  fluoresce 
brilliantly.  These  streams  of  electrons  are  called  cathode  rays. 

The  fact  that  they  possess  a  negative  charge  may  be  demon- 
strated by  placing  two  parallel  plates  within  the  tube  between 
which  there  exists  a  difference  of  potential.  A  stream  of  cathode 
rays  passed  between  them  will  be  deflected  away  from  the  nega- 
tively charged  plate  toward  the  positive.  Again,  if  a  magnetic 
field  is  introduced  across  the  tube,  the  stream  will  be  deflected 


CONDUCTION  THROUGH  GASES 


209 


at  right  angles  both  to  their  motion  and  to  the  field  in  the  manner 
required  by  the  ordinary  rules  of  electrodynamic  action  for 
currents. 

160.  Velocity  and  Ratio  of  the  Charge  to  the  Mass  of  an 
Electron.1 — The  fact  that  an  electron,  when  moving  through  a 
magnetic  field,  is  acted  upon  by  a  force  at  right  angles  both  to  its 
motion  and  the  direction  of  the  field  may  be  used  to  determine 
the  ratio  of  the  charge  to  the  mass  of  an  electron  and  the  velocity 
with  which  it  moves.  Apparatus  arranged  for  this  purpose  is 
shown  in  Fig.  114.  A  vacuum  chamber  C  is  constructed  from  a 
brass  tube  from  which  there  projects  a  smaller  tube  A  also  of 


FIG.  114. — Apparatus  for  measuring 


brass.  The  end  of  A  is  tapered  and  fitted  to  one  end  of  a  ground 
glass  joint.  The  other  end  of  the  glass  tube  is  closed  and  carries 
the  cathode  K.  A  piece  of  plate  glass  P,  on  the  inner  side  of 
which  has  been  placed  a  thin  coating  of  fluorescent  material 
such  as  calcium  tungstate  closes  the  vacuum  chamber.  The  end 
of  the  smaller  tube  A  contains  a  brass  plug  through  which  has 
been  bored,  with  a  jeweler's  drill,  a  very  fine  hole. 

When  a  suitable  vacuum  has  been  obtained,  a  discharge  pro- 
duced between  A  and  K  by  a  static  machine  M  causes  a  stream 
of  electrons  to  pass  from  K  to  A ,  the  individual  electrons  of  which 
move  in  straight  lines  normal  to  K.  All  but  those  lying  in  a 

1  TOWNSEND,  Electricity  in  Gases,  p.  453. 
CROWTHER,  Ions,  Electrons  and  Ionizing  Radiations,  p.  92. 
DUFF,  A  Text  Book  of  Physics,  p.  492. 
14 


210  ELECTRICITY  AND  MAGNETISM 

very  narrow  beam,  defined  by  the  hole  through  A,  are  stopped 
but  those  passing  through,  enter  the  chamber  C  and  produce  on 
P  a  bright  fluorescent  spot.  If  now  the  solenoid  is  energized, 
the  magnetic  field  causes  a  deflection  of  the  beam  and  the  spot 
is  moved  a  distance  d  perpendicular  to  the  plane  of  the  paper, 
(shown  in  the  plane  of  the  paper  in  Fig.  114). 

If  the  magnetic  field  is  uniform,  the  path  of  an  electron  is 
circular,  since  the  force,  in  this  case,  is  constant  in  magnitude, 
and  is  always  at  right  angles  to  the  motion.  The  magnitude  of 
the  force  may  be  obtained  as  follows:  Let  I  be  the  length  of  path 
of  an  electron  through  the  magnetic  field.  When  it  has  traversed 
the  distance  I,  a  quantity  of  electricity  e  has  been  transported 

through  this  distance  and  may  be  replaced  by  a  steady  current  of 

g 
strength  i  defined  by  i  =  -,  where  t  is  the  time  required  for  the 

electron  to  travel  the  distance  I.  The  theory  of  electrodynamics 
gives,  for  the  force  acting  on  a  conductor  of  length  I,  carrying  a 
current  i,  the  expression 

F  =  Hil  =  Hel  =  Hev  ^ 

where  v  is  the  velocity  with  which  the  electron  moves. 

Since  the  electron  moves  in  a  circle,  whose  radius  we  will  call 
R,  the  force  given  by  eq.  (1)  must  balance  the  centrifugal  force. 
Accordingly,  we  have 

'         >     (2)' 


where  m  is  the  mass  of  the  electron.  The  velocity  v  is  acquired 
while  the  electron  moves  through  the  difference  of  potential  E 
maintained  between  the  anode  and  cathode  by  the  static  machine. 
Since  there  is  no  potential  difference  between  A  and  P  it  travels 
this  distance  with  constant  velocity.  The  kinetic  energy 
acquired  in  moving  from  K  to  A  is  equal  to  the  loss  of  potential 
energy  over  this  distance.  From  the  law  of  conservation  of 
energy  and  the  definition  of  potential  difference,  we  have 

Ee  =  Y2mv*  (3) 

p 

Eliminating  successively  v  and     L  from  eqs.   (2)  and  (3)  there 

Tit 

results 

e         2E  2E  . 


CONDUCTION  THROUGH  GASES  211 

The  radius  of  curvature  R  is  obtained  from  the  sagitta  formula 

*=£  <«> 

The  magnetic  field  strength  H  is  computed  from  the  dimensions 
of  the  solenoid  and  the  current  through  it  by  the  formula 

" 


where  N  is  the  number  of  turns  on  the  solenoid  and  L  its  length. 
The  accelerating  potential  E  is  measured  by  an  electrostatic 

voltmeter. 

0 
161.  Experiment   33.     Measurement   of  —    and    Velocity  for 

an  Electron  in  a  Cathode  Ray.  —  Connect  the  apparatus  as  shown 
in  Fig.  114.  Start  the  static  machine  and  pump  the  vacuum 
chamber  until  a  green  fluorescence  is  seen  near  the  anode  A. 
Should  this  color  appear  at  K  the  leads  to  the  static  machine 
should  be  reversed.  A  bright  spot  will  appear  at  P.  Energize 
the  solenoid  and  determine  the  current  required  for  a  suitable 
deflection  d.  In  taking  observations,  reverse  the  solenoid  current 
and  measure  2d.  It  will  be  found  that  by  varying  the  vacuum, 
different  voltages  may  be  maintained  across  the  discharge  while 
the  static  machine  is  driven  at  a  constant  speed.  With  the 
two  halves  of  the  solenoid  as  close  together  as  possible,  take  a 
series  of  observations  using  different  accelerating  voltages,  and 

a 

deflecting  fields  and  determine  —  and  v.     The  fact  that  the  parts 

of  the  solenoid  must  be  separated  to  permit  the  entrance  of  the 
discharge  tube  introduces  a  non-uniformity  in  the  field.  To 
determine  this  error,  take  a  series  of  observations,  keeping  the 
accelerating  potential  and  the  solenoid  current  constant  and 
increase  the  separation  of  the  solenoid  parts  from  the  smallest 

p 

amount  up  to  10  cms.,  and  plot  the  apparent  values  of  —  and  v 

as  a  function  of  the  separation.  The  intercept  of  this  curve, 
when  extrapolated  to  zero  separation  gives  the  correction  to  be 

p 

applied  to  the  results  obtained  above.     Since  the  value  of  — 

is  usually  given  in  electromagnetic  units  per  gram,  it  is  necessary 
to  express  E  and  H  in  eq.  (4)  in  that  system. 

Report.  —  Plot  the  correction  curve  called  for  above  and  apply 

/> 

to  average  values  of  —  and  v.     Compute  the  velocity  of  an  elec- 


212  ELECTRICITY  AND  MAGNETISM 

iron  which  has  fallen  through  the  following  differences  of  poten- 
tial using  your  value  for—:  300,  3,000,  30,000  volts.  Compute 

the  time  required  for  an  electron  to  move  from  K  to  A  for  some 
one  of  the  conditions  actually  used  in  this  experiment.  If  the 
charge  on  an  electron  is  4.77  X  10~10  electrostatic  units,  compute 
the  number  of  electrons  passing  per  second  across  a  plane  in  a 
wire  through  which  a  current  of  one  ampere  is  flowing. 

162.  Radio-active  Substances.1 — If  the  region  surrounding  any 
radio-active  substance  such  as  uranium,  radium,  thorium,  etc., 
is  examined  by  appropriate  means,  it  is  found  that  these  sub- 
stances emit  definite  radiations  which  have  very  unusual  prop- 
erties.    These  radiations,   for  example,   are  able  to  darken  a 
photographic  plate,  to  convert  an  insulating  gas  into  a  conductor, 
and  to  cause  a  fluorescent  screen  to  emit  light.     Moreover,  they 
are  different  from  ordinary  light  in  that  they  are  able  to  pene- 
trate many  substances  usually  regarded  as  opaque.     It  has  been 
found  that  each  radio-active  substance  is  a  definite  chemical 
element  and  that  its  activity  is  due  to  a  spontaneous  decomposi- 
tion or  disintegration  of  its  atoms.     Furthermore,  when  certain 
of  the  rays  are  emitted,  there  is  a  definite  reduction  in  the  atomic 
weight  of  the  substance,  which  naturally  leads  to  the  view  that 
the  atoms  of  these  substances  are  made  up  of  complex  systems 
which  have  the  same  intrinsic  character  and  differ  from  one 
another  only  in  their  order  of  arrangement  or  degree  of  complex- 
ity.    Three  distinct  types  of  radiation  have  been  found  which 
are  designated  as  a,  0,  and  7  rays. 

163.  The  Alpha  Rays. — These  rays  are  distinguished  from  the 
others  by  the  fact  that  they  are  easily  absorbed  on  passing 
through  gases  or  thin  sheets  of  metal  and  that  their  action  on  a 
photographic  plate  is  weak.     On  the  other  hand,  they  are  very 
effective  as  a  means  for  ionizing  a  gas,  and  they  cause  fluorescent 
substances  to  emit  light.     If  a  screen  upon  which  they  are 
acting  is  examined  by  a  microscope,  it  is  found  that  the  illumina- 
tion is  not  uniform  but  is  made  up  of  a  large  number  of  separate 
flashes   as   though   the   screen   were   under  bombardment.     In 
fact  it  has  been  found  that  a  rays  are  discrete  particles  shot  out 

1  CROWTHER,  Ions,  Electrons  and  Ionizing  Radiations,  chap.  XI. 

McCLUNG,  Conduction  of  Electricity  through  Gases  and  Radioactivity. 
Part  II. 

DUFF,  A  Text  Book  of  Physics,  p.  502. 


CONDUCTION  THROUGH  GASES  213 

from  radio-active  substances  and  it  is  possible  by  suitable 
experimental  arrangements,  to  photograph  their  zig-zag  courses 
as  they  make  their  way  through  a  gas,  abruptly  deflected  by 
some  of  the  gas  molecules,  and  stopped  by  others. 

If  a  beam  of  a  particles  is  shot  at  right  angles  to  an  electric 
or  a  magnetic  field,  the  path  js  curved  in  much  the  same  manner 
as  the  cathode  ray  stream  described  above,  except  that  the 
deflection  is  much  smaller  in  magnitude  due  to  their  larger  mass 
and  is  in  the  opposite  direction,  indicating  that  they  are  posi- 
tively charged.  By  making  use  of  electric  and  magnetic  deflec- 
Q 

tions.  the  value  —  of  the  ratio  of  the  charge  to  their  mass  and  the 
m 

velocity  with  which  they  are  emitted,  have  been  measured.     The 

a 

results  show  that  —  is  the  same  for  all  a.  particles,  no  matter 
m 

what  their  source  and  is  equal  to  4,823  electromagnetic  units  per 
gram,  and  that  the  velocities  range  from  1.5  X  109  to  2.2  X  109 
cms.  per  sec. 

The  ratio  of  the  charge  to  the  mass  for  the  hydrogen  ion  in 
electrolysis  is  twice  that  for  the  a  particle,  and  at  first  sight 
it  might  be  supposed  that  the  latter  is  a  hydrogen  molecule 
consisting  of  two  atoms.  However,  it  has  been  found  that  the 
charge  carried  by  the  a.  particle  is  twice  that  of  the  hydrogen 
ion,  and  hence  its  mass  must  be  four  times  that  of  the  hydrogen 
atom.  Since  the  particle  is  atomic  in  size  and  is  of  the  same 
order  of  magnitude  as  the  atom  of  helium  whose  atomic  weight 
is  3.96,  the  most  natural  assumption  is  that  it  is  an  atom  of  helium 
with  twice  the  electric  charge  of  the  hydrogen  ion.  This  hypo- 
thesis is  supported  by  the  fact  that  both  chemical  and  spectro- 
scopic  analyses  show  conclusively  that  helium  is  always  present 
where  radio-active  transformations  are  taking  place. 

164.  The  Beta  Rays. — The  0  rays  are  distinguished  from  a  rays 
in  several  important  respects.  In  the  first  place,  they  have 
a  far  greater  penetrating  power.  While  the  a  rays  are  completely 
stopped  by  a  sheet  of  aluminum  foil  Jf  o  mm.  in  thickness,  /?  rays 
still  produce  noticeable  effects  after  passing  through  sheets 
100  times  this  thickness.  Again,  they  are  much  more  easily 
deflected  by  a  magnetic  field.  The  deflection  of  the  a.  rays  is 
appreciable  only  in  the  largest  fields  available,  and  even  then 
special  methods  have  to  be  employed.  The  j8  particles,  on  the 
other  hand,  travel  in  circles  of  large  curvature  when  moving  at 


214  ELECTRICITY  AND  MAGNETISM 

right  angles  to  fields  of  ordinary  magnitudes.  The  direction  of 
the  deflection  shows  that  they  carry  a  negative  charge,  and  all 
the  evidence  indicates  that  they  are  identical  with  the  cathode 
rays  of  the  ordinary  discharge  tube,  i.e.,  electrons. 

By  subjecting  /?  rays  to  the  deflecting  action  of  electric  and 

& 

magnetic  fields  combined,  the  values  of  --  and  the    velocities 

with  which  they  are  emitted  may  be  measured.  It  has  been 
found  that  while  the  former  is  the  same  as  for  the  cathode-ray 
particles,  the  velocities  of  emission  are  considerably  higher  than 
those  observed  in  discharge  tubes,  ranging  from  6  X  109  to 
2.8  X  1010  cms.  per  second.  The  latter  is  very  close  to  the 
velocity  of  light,  3  X  1010  cms.  per  second. 

o 

A  careful  study  has  been  made  by  Kaufmann  of  the  value  —  for 

wi> 

o 

the  particles  as  a  function  of  velocity,  and  it  was  found  that  —  is 

not  constant,  but  decreases  as  the  speed  increases.  This  can  be 
explained  only  by  assuming  that  e  decreases  or  that  m  increases 
as  the  velocity  becomes  larger.  The  evidence  furnished  by  other 
lines  of  study  indicates  that  the  charge  of  the  electron  is  one  of  the 
fixed  constants  of  nature,  and  therefore  it  is  concluded  that  the 
mass  of  the  electron  depends  upon  its  velocity.  Theoretical 
considerations  have  shown  that  the  apparent  mass  of  an  electron 
is  due  wholly,  or  in  part,  to  the  motion  of  its  electric  charge.  In 
fact,  for  a  number  of  years,  the  view  was  held  that  the  mass  of 
the  electron  is  entirely  electromagnetic  in  character,  but  some 
very  recent  work  indicates  that  this  can  not  be  the  case  entirely. 

165.  The  Gamma  Rays. — The  nature  of  the  7  rays  is  very 
different  from  that  of  the  a  and  /?  rays.  They  are  distinguished 
by  the  fact  that  they  possess  very  much  greater  power  of  pene- 
tration. In  fact  they  may  easily  be  detected  after  passing  through 
several  cms.  of  iron.  Though  subjected  to  the  most  powerful 
electric  and  magnetic  fields  available,  they  show  no  deflection, 
and  can  not  therefore  carry  an  electric  charge.  They  cause  a 
fluorescent  screen  to  emit  light,  and  affect  a  photographic  plate. 
When  passed  through  gases  they  produce  ionization,  and,  in 
fact,  are  usually  detected  by  this  action. 

Searching  investigations  have  shown  that  they  are  similar  in 
character  to  X-rays,  that  is,  electromagnetic  waves  of  very  short 
wave  length.  The  similarity  of  the  relation  of  0  rays  to  cathode 


CONDUCTION  THROUGH  GASES  215 

rays  and  7  rays  to  X-rays  is  very  close.  When  the  target  of  an 
X-ray  tube  is  struck  by  a  rapidly  moving  electron,  the  electronic 
orbits  of  one  of  the  atoms  of  the  former  undergoes  some  sort  of 
rearrangement ;  that  is,  they  change  over  from  one  stable  config- 
uration to  another  possessing  a  different  amount  of  potential 
energy,  and  a  train  of  X-raysls  emitted.  The  emission  of  the  X-ray 
occurs  as  the  result  ,pf  suddenly  stopping  a  high  speed  electron. 
Similarly,  when  a  radio-active  substance  emits  a  0  particle,  send- 
ing it  forth  with  a  velocity  comparable  to  that  of  light,  a 
rearrangement  of  the  electronic  orbits  also  occurs,  which  is  accom- 
panied by  the  emission  of  the  7  ray.  The  7  rays  thus  accompany 
the  rapid  acceleration  of  electrons.  The  fact  that  7  rays  are 
always  present  when  /?  rays  are  emitted  supports  this  view. 
Measurements  have  shown  that  the  wave  length  of  the  7  rays 
is  somewhat  shorter  than  that  of  the  most  penetrating  X-rays. 

166.  Radio-active  Transformations. — Careful  investigations 
of  the  phenomena  accompanying  the  emission  of  the  rays  just 
described,  show  that  radio-active  substances  are  distinguished 
from  ordinary  ones  in  that  they  are  constantly  undergoing  changes 
of  character,  never  observed  in  ordinary  materials.  Each  sub- 
stance is  entirely  distinct  from  the  other,  and  has  its  own  charac- 
teristic physical  and  chemical  properties.  However,  instead  of 
enduring  indefinitely  as  is  the  case  with  ordinary  elements,  such 
as  copper,  iron,  gold,  etc.,  each  radio  active  substance  has  a  definite, 
measurable  period  of  existence,  after  which  it  disintegrates  and 
becomes  a  new  chemical  substance,  and  it  is  during  these  proc- 
esses of  transformation,  that  the  emission  of  rays  occurs. 

All  molecules  are  made  up  of  atoms  which  consist  of  positive 
nuclei  with  electrons  rotating  about  them  in  closed  orbits.  The 
electrons  are  held  in  their  orbits  by  the  electric  attractions 
existing  between  them  and  the  nucleus  while  the  atoms  are  held 
together  by  the  electric  forces  between  their  parts,  or  the  magnetic 
forces  due  to  the  circulating  electrons.  This  complicated  struc- 
ture becomes  unstable  for  some  reason  or  another,  and  an  a 
or  a  |8  particle  or  both  is  emitted.  After  a  rearrangement  of  the 
remaining  particles,  a  new  state  of  stable  equilibrium  ensues, 
giving  a  new  substance  of  different  physical  and  chemical  pro- 
perties. As  an  illustration,  take  the  substance  radium.  Although 
the  individual  molecules  do  not  have  the  same  periods  of  existence, 
the  life  of  an  average  molecule  is  2,000  years.  At  the  end  of  this 
time,  it  emits  an  a  particle,  and  the  residue  is  called  radium 


216  ELECTRICITY  AND  MAGNETISM 

emanation.  The  emanation  persists  for  a  period  of  3.75  days 
when  it  gives  off  another  a  particle,  and  becomes  radium  A .  In 
this  form  it  lasts  for  3  minutes,  then  again  emits  an  a.  particle  and 
becomes  radium  B.  This  state  persists  for  26  minutes  when  it 
gives  off  a  jS  particle  accompanied  by  a  7  ray  and  becomes  radium 
C,  and  so  on.  The  entire  series  has  been  carefully  worked  out, 
starting  with  uranium,  going  through  ionium  and  the  various 
phases  of  radium,  and  thorium  to  those  of  actinium.  The  duration 
of  the  different  phases  ranges  from  a  few  minutes  to  1010  years. 
Some  of  the  transformations  are  apparently  not  accompanied  by 
the  emission  of  any  rays.  These  transformations  are  explained  by 
supposing  that  the  ray  is  present  but  possesses  such  a  low  velocity 
as  to  be  unable  to  ionize  a  gas  and  is  therefore  not  detected. 

It  is  important  to  note  that  each  time  an  a  particle  is  emitted 
the  atomic  weight  decreases  by  4,  i.e.,  the  atomic  weight  of  helium. 
Furthermore,  the  last  radio  active  product,  radium  F,  or  polon- 
ium, has  an  atomic  weight  equal  to  that  of  lead,  and  possesses 
the  properties  of  ordinary  lead. 

It  is  easy  to  conjecture  that  each  of  the  chemical  elements  as 
we  know  them,  has  been  derived  from  one  higher  in  the  scale  of 
atomic  weights  by  the  emission  of  one  or  more  a  particles,  and 
that  transformations  are  going  on  continuously  but  at  a  rate  so 
slow  as  to  escape  detection  by  methods  at  present  available. 

167.  Experiment  34.  lonization  by  Radio-active  Substances. — 
The  apparatus  for  this  experiment  is  shown  in  Fig.  115.  It 
consists  of  an  ionization  chamber  made  entirely  of  metal.  The 
radio-active  substance,  in  the  form  of  a  powder  is  spread  over  the 
plate  A  which  may  be  moved  up  or  down.  An  insulated  plate  D 
is  connected  to  an  electrometer  E  mounted  in  another  chamber 
and  connected  with  the  ionization  chamber  by  a  removable  brass 
tube.  The  electrometer  is  charged  by  means  of  a  battery  B  of 
small  dry  cells  by  pressing  down  the  wire  W  which  must  be 
insulated  from  the  container.  When  the  rays  from  the  radio-active 
substance  pass  up  through  the  metal  gauze  G  they  ionize  the  air 
between  G  and  D.  Either  electrons  or  positive  ions,  depending 
upon  the  sign  of  the  charge  on  D  and  E  are  drawn  toward  D  and 
neutralize  this  charge.  The  deflections  of  the  electrometer  are 
read  by  means  of  a  long  focus  microscope  provided  with  an  eye 
piece  having  a  graduated  scale.  The  time  required  for  the  gold 
leaf  of  the  electrometer  to  fall  through  one  division  is  inversely 
proportional  to  the  ionization  current. 


CONDUCTION  THROUGH  GASES 


217 


It  is  necessary  first  to  determine  the  rate  of  discharge  of  the 
ionization  chamber  and  electrometer  due  to  leakage  alone.  For 
this  purpose,  cover  up  the  radio  active  substance  by  a  close  fitting 
metallic  plate  P.  Charge  the  electrometer  by  connecting  it  for 
an  instant  to  the  battery  by  means  of  W,  and  note  the  time 
required  for  the  gold  leaf  to  fall  one  division.  Take  several 
readings  and  average.  Thi&  leakage  is  due  partly  to  imper- 
fect insulation,  and  partly  to  7  rays  which  penetrate  the  metal 
cover. 

Remove  the  shield  P,  charge  the  electrometer  as  before,  and 
with  A  near  the  bottom  of  the  chamber,  determine  the  rate  of 

w 


D 


1 


FIG.  115. — Apparatus  for  ionization  studies. 

discharge  as  above.  The  difference  between  these  two  rates  is 
a  measure  of  the  ionization  produced  by  the  /3  rays  from  the  radio 
active  substance,  provided  the  distance  AC  is  greater  than  10 
centimeters,  the  range  of  the  a  particles.  Take  a  series  of 
observations  determining  the  rate  of  leak  each  time  raising  A  % 
cm.  until  a  marked  increase  in  the  rate  of  leak  is  observed.  This 
indicates  that  the  a  rays  have  penetrated  the  space  between  the 
gauze  and  D.  Take  readings  each  millimeter  until  the  rate  has 
become  nearly  constant  again.  Continue  until  the  plate  A  is 
as  high  as  it  can  be  raised. 

Report. — Plot  inverse  time  of  leakage  against  distance  between 
A  and  D.  Draw  horizontal  lines  representing  the  leakage  cur- 
rent, and  that  due  to  the  currents  produced  by  both  $  and  a.  ray 
ionization. 


CHAPTER  XIV 
ELECTRON  TUBES1 

During  the  last  decade,  the  electron  tube  has  had  a  develop- 
ment little  less  than  phenomenal.  Because  of  the  multiplicity 
of  its  uses,  e.g.,  as  detector,  amplifier,  oscillator,  modulator,  etc., 
it  finds  many  applications  not  only  in  the  art  of  radio  communi- 
cation but  also  in  engineering  work  and  the  general  research 
laboratory.  No  student  of  electrical  engineering  or  physics  'can 
afford  to  be  unacquainted  with  this  device,  and  it  is  the  purpose 
of  the  present  chapter  to  set  forth  and  illustrate  the  fundamental 
principles  upon  which  it  operates. 

168.  Free  Electrons. — It  is  customary  to  distinguish  an  insu- 
lator from  a  conductor  by  saying  that  in  the  former,  electrons 
are  held  in  their  orbits  about  the  positive  nuclei  with  forces  so 
great  that  they  can  be  dislodged,  if  at  all,  only  by  exceedingly 
large  fields  while  in  the  latter,  the  attracting  forces  are  so  weak 
that  they  are  easily  torn  from  their  positions  of  equilibrium  and 
move  about  through  the  body.  The  idea  of  such  easily  disrup- 
table  atoms  is  held  by  some  to  be  inconsistent  with  the  rigid 
mechanical  properties  qf  metals,  and  the  ease  with  which  elec- 
trons move  through  conductors  is  explained  by  saying  that  while 
the  forces  holding  atoms  together  are  very  large,  nevertheless, 
due  to  the  closeness  of  approach  during  collisions,  the  nucleus  of 
one  atom  may  attract  an  electron  of  another  with  so  great  a  force 
in  the  opposite  direction  that  the  electron  is  nearly  in  equilib- 
rium, and  a  slight  field  may  cause  it  to  leave  its  original  atomic 
system  and  enter  that  of  a  neighboring  one.  The  one  from  which 
it  escaped  would  be  left  with  an  excess  positive  charge  and  might, 
in  a  similar  manner,  capture  an  electron  from  another  atom. 
According  to  this  view,  electrons  move  through  a  conductor,  not 
by  zig-zag  paths  between  molecules,  but  by  passing  through 
the  molecules,  and  forming  distinct  parts  of  the  atomic  structures 

1  RICHARDSON,  Emission  of  Electricity  from  Hot  Bodies. 
VAN  DER  BIJL,  Thermionic  Vacuum  Tube. 
MORECROFT,  Principles  of  Radio  Communication. 
LAUER  and  BROWN,  Radio  Engineering  Principles. 

218 


ELECTRON  TUBES  219 

on  the  way.  Because  of  the  many  collisions  taking  place  in 
consequence  of  thermal  agitation,  the  large  number  of  electrons 
required  to  explain  observed  currents  is  easily  accounted  for. 

169.  Electron  Emission. — From  the  fact  that  electrons  move 
thus  freely  from  one  part  of  a  conductor  to  another,  going  either 
between  the  molecules  or  through  them,  and  pass  readily  out  of 
one  conductor  in  to  another  it!  contact  with  it,  it  might  be  inferred 
that  they  could  also  be  drawn  easily  out  of  a  conductor  into  a 
vacuuous  space.  It  is  found,  however,  that  this  is  not  the  case, 
and  that  special  means  must  be  used  to  cause  them  to  thus 
emerge.  For  want  of  a  better  explanation,  it  has  been  assumed 
that  there  exists  at  the  surface  of  a  conductor,  a  force  which 
tends  to  keep  the  electrons  within  the  body.  The  exact  nature 
of  this  force  is  unknown,  but  recent  developments  regarding  the 
structure  of  the  atom  tend  to  support  the  view  that  such  a  force 
really  exists.  If  this  is  true,  a  certain  amount  of  work  must  be 
done  on  the  electron  to  move  it  out  of  a  body  against  this  attract- 
ing force. 

One  of  the  ways  in  which  electrons  may  be  dislodged  from  a 
metal  is  by  the  application  of  electromagnetic  radiation  of  very 
short  wave  length.  For  example,  if  ultraviolet  light  falls  upon 
an  insulated  piece  of  zinc,  it  acquires  a  positive  charge,  or,  if 
originally  charged  to  a  negative  potential,  it  loses  this  charge 
and  becomes  positive.  This  is  explained  by  saying  that  the 
electrons  within  the  atoms  of  the  metal  absorb  energy  from  the 
incident  light  waves  and  are  stimulated  to  vibrate  with  ampli- 
tudes so  great  that  they  possess  energy  sufficient  to  overcome 
the  surface  forces  and  escape  into  the  surrounding  space  with 
velocities  which  depend  upon  the  energy  of  the  light  wave  and 
that  lost  in  moving  against  the  surface  attraction.  This  is 
known  as  the  photo  electric  effect,  and  while  it  is  most  pronounced 
in  zinc  it  is  found  to  exist  to  a  greater  or  less  extent  in  all  metals. 
Another  way  in  which  electrons  may  be  dislodged  from  a  body 
is  by  bombardment  with  other  electrons.  Certain  metals, 
notably  copper  and  nickel,  when  struck  by  electrons  having 
sufficient  energy  of  motion  may  emit  as  many  as  twenty  other 
electrons  for  each  one  striking.  This  is  known  as  secondary 
emission,  and  has  been  made  use  of  in  a  number  of  electron 
devices. 

The  most  effective  way  to  get  electrons  out  of  a  body  is  to 
heat  it.  The  explanation  of  this  effect  is  as  follows:  Since  the 


220  ELECTRICITY  AND  MAGNETISM 

body  possesses  temperature,  its  molecules  must  be  in  motion 
and  the  average  kinetic  energy  of  the  molecules  is  a  measure  of 
the  temperature.  The  electrons  being  free  to  move  about  within 
the  body  must  also  possess  undirected  motion  of  thermal  agita- 
tion from  impact  with  the  molecules.  In  fact  it  is  generally 
supposed  that  they  are  in  thermal  equilibrium  with  the  molecules, 
that  is,  the  average  value  of  their  kinetic  energies  is  the  same 
as  that  of  the  molecules.  Since  kinetic  energy  is  J^  mv2,  and  the 
mass  of  an  electron  is  very  much  less  than  that  of  a  molecule,  it 
follows  that  the  velocities  of  the  electrons  must  be  many  times 
larger  than  those  of  the  molecules.  The  temperature  accord- 
ingly need  not  be  very  high  (dull  red)  before  an  appreciable 
number  of  electrons  will  possess  sufficient  energy  of  motion  to 
overcome  the  surface  force  of  attraction  and  escape  into  the 
surrounding  space.  If  the  emitting  body  is  insulated,  it  will 
take  on  a  positive  charge  because  of  the  loss  of  electrons.  If  the 
body  is  in  a  closed  vessel  so  that  the  electrons  can  not  move  far 
away,  some  of  them  will  be  drawn  back  into  it,  and  an  equilib- 
rium condition  will  be  established  in  which  the  number  emitted 
is  equal  to  the  number  falling  back.  The  number  of  electrons 
emitted  per  unit  time  is  given  by  the  formula1 

_  ^ 
N  =  A\/Te     T 

where  T  is  the  absolute  temperature  of  the  body,  e,  the  base  of  the 
Naperian  logarithms,  and  A  and  b  are  constants  depending  upon 
the  nature  of  the  substance,  its  size,  shape,  and  certain  other 
characteristics. 

170.  The  Two -element  Electron  Tube. — This  is  a  device  in 
which  application  of  thermionic  emission  is  made  for  the  rectifica- 
tion and  control  of  currents.  It  consists  of  a  filament  F  which 
may  be  made  of  tungsten  or  of  platinum  coated  with  oxides  of 
barium,  stromtium  or  calcium  and  a  plate  P  mounted  within  an 
enclosure  which  has  been  exhausted  to  a  very  high  vacuum.  The 
filament  may  be  heated  to  any  desired  temperature  by  a  battery 
A  as  shown  in  Fig.  116.  Another  battery  B  is  connected  between 
the  plate  and  the  filament  in  such  a  way  as  to  make  P  positive 
with  respect  to  F.  On  heating  the  filament,  electrons  pass  out 
into  the  enclosure,  and,  were  it  not  for  the  battery  B,  would  fill 
the  space  with  a  definite  density  depending  upon  the  temperature 

1  RICHARDSON,  Phil.  Trans.  (A)  vol.  201,  1903,  p.  543. 


ELECTRON  TUBES 


221 


FIG.   116. — Two  element 
electron  tube. 


of  the  filament,  and  an  equilibrium  state  would  be  reached  in 

which  the  number  returning  to  the  filament  is  equal  to  that  of 

those  leaving  it.     The  battery  B  produces  an  electric  field  which 

causes   electrons   to   move   from   the 

filament    to    the    plate,    which    they 

enter  and  pass  through  the. external 

metallic    circuit    and    return   to  the 

filament.     They  constitute  an  electric 

current  which,  according  to  common 

parlance,  is  said  to  flow  into  the  plate 

and  out  of  the  filament.     The  passage 

of  electrons  through  the  vacuous  space 

between  the  electrodes  is  called  the 

"space  current."     The  magnitude  of 

the  space  current  is  limited  by  two  important  considerations 

which  may  be  made  clear  by  the  following  experiments. 

171.  Voltage    Saturation. — Suppose    that    the    temperature 

of  the  filament  is  held  constant  at  a  value  somewhat  less  than  that 

required  for  normal  operation  of  the  tube.     Let  the  voltage  of  the 

battery  B  be  gradually  increased  from  zero  to  some  specified 

value,  and  let  the  space  current  be  measured  and  plotted  as  a 

function  of  B.  It  will  be 
found  that  for  small  values 
of  plate  potential  the  space 
current  gradually  increases 
as  shown  by  the  curve  of 
Fig.  117,  marked  TI,  but  that 
it  soon  stops  rising  and  re- 
mains constant,  no  matter 
how  much  the  voltage  is 
increased.  This  limitation  of 
the  current  is  due  to  the  fact 
that  there  is  available  at  the 
filament  only  a  finite  number 
of  electrons  which  depends 


Plate  Potential 
FIG.  117. — Voltage  saturation  curves. 

upon      its     temperature     as 

shown  by  eq.  (1).  When  the  voltage  is  sufficient  to  draw  to 
P  all  the  electrons  which  are  emitted  at  a  given  temperature, 
the  maximum  current  for  that  temperature  is  reached,  and  no 
increase  in  voltage,  however  great,  can  further  increase  the  cur- 
rent. If  now,  the  experiment  is  repeated  using  a  higher  filament 


222  ELECTRICITY  AND  MAGNETISM 

temperature,  the  lower  part  of  the  curve  will  be  the  same  as  in 
the  previous  case.  When  such  a  voltage  is  reached  that  the 
available  electrons  are  all  drawn  to  the  plate,  the  current  again 
becomes  constant  but  this  time  at  a  higher  value,  as  shown  by 
Tz  of  the  figure.  In  this  way  a  series  of  curves  may  be 
obtained  which  are  coincident  at  their  lower  extremities  but 
become  horizontal  at  definite  values  of  voltage  for  each  filament 
temperature. 

The  constant  current  which  results  when  all  the  available 
electrons  are  used  is  somewhat  inappropriately  called  the  "  satura- 
tion current"  from  the  similarity  of  shape  between  these  curves 
and  the  magnetization  curves  of  iron,  in  which  the  knee  of  the 
curve  is  called  the  saturation  point.  The  voltage  required  to 
produce  the  saturation  current  at  any  temperature  is  called  the 
" saturation  voltage"  for  that  temperature.  The  saturation 
current  is  thus  a  measure  of  the  total  electron  emission  at  a  given 
temperature. 

172.  Space  Charge. — From  the  experiment  just  described,  it 
might  be  inferred  that  the  rate  of  electron  emission  is  the  only 

limitation  to  the  magnitude  of 
the  space  current  and  that  if 
filaments  of  sufficient  areas  were 
provided,  currents  of  any  magni- 
tude could  be  obtained.  That 
this  is  not  the  case  is  shown  by 
the  following  experiment. 

Suppose  now  the  voltage  of 
the  battery  B  is  held  constant 
and  the  space  current  measured 
as  the  temperature  of  the  fila- 
ment is  changed.  Starting  with 


Filament  Temperature  ,,          «i  -,  -,       .,          -ni 

the    filament    cold,    it    will    be 

of  space  charge.       foun<i   ^   ^  gp&ce  ^^  ;g 

zero,  since  no  electrons  are  emitted.  In  fact  for  most  filament 
materials,  there  will  be  no  space  current,  measurable  with 
ordinary  instruments,  until  the  filament  is  hot  enough  to  show 
a  dull  glow.  When  this  point  has  been  reached,  it  is  found 
that  the  space  current  rises  rather  rapidly  with  increasing  fila- 
ment temperature.  This  increase  in  space  current  does  not 
continue  indefinitely  for  a  temperature  is  soon  reached  above 
which  the  space  current  remains  constant  as  shown  by  the  curve 


ELECTRON  TUBES  223 

marked  V\  of  Fig.  118.  Even  though  the  temperature  of  the 
filament  is  raised  to  the  melting  point,  the  space  current  remains 
constant.  If,  however,  the  experiment  is  repeated  using  a  larger 
voltage  for  the  battery  B,  it  is  found,  on  starting  again  with  the 
filament  cold,  that  the  relation  between  the  space  current  and 
filament  temperature  is  the  same  as  in  the  previous  case  for  low 
temperatures.  At  a  certain*'  temperature,  however,  the  space 
current  again  becomes  constant  but  has  a  larger  value  than 
before,  and  takes  place  at  a  higher  filament  temperature.  This 
is  shown  by  the  curve  F2  of  the  figure.  Repeating  with  a  still 
higher  voltage,  the  curve  F3  is  obtained. 

The  limitation  of  the  space  current  in  this  case  is  due  to  the 
action  of  the  electrons  which  constitute  it.  Consider  an  elec- 
tron which  has  just  emerged  from  the  filament.  If  it  were  the 
only  electron  between  the  filament  and  the  plate,  it  would  be 
acted  upon  by  an  electric  field  which  depends  only  upon  the  dif- 
erence  of  potential  between  the  filament  and  the  plate.  If  how- 
ever, there  exists  between  this  electron  and  the  plate  a  second 
electron,  the  force  on  the  first  electron  will  be  less  than  in  the  pre- 
vious case  since  it  is  repelled  by  the  second  electron.  The  fact 
that  the  second  electron  may  be  in  motion  makes  no  difference  so 
long  as  its  velocity  is  less  than  that  of  light.  If,  now,  there  is  a 
swarm  of  electrons  of  sufficient  number  between  the  filament 
and  plate,  their  repelling  action  on  freshly  emitted  electrons  will 
just  balance  the  attraction  of  the  positively  charged  plate  and  there 
is  no  tendency  for  them  to  move,  until  some  of  those  near  the  plate 
have  entered  it  and  thus  reduced  the  number  in  the  swarm. 
Electrons  from  the  filament  will  then  enter  the  swarm  keeping  the 
number  between  filament  and  plate  constant,  thus  giving  the  steady 
space  current  observed.  If  the  plate  voltage  is  raised,  it  will  re- 
quire a  larger  number  of  electrons  in  the  space  to  neutralize  this  in- 
creased potential  gradient.  Furthermore,  electrons  will  be  drawn 
out  of  the  swarm  more  rapidly  thus  requiring  a  larger  number  to 
enter  it  to  maintain  equilibrium  and  the  space  current  is 
thereby  increased. 

From  the  explanation  just  given,  it  may  be  inferred  that  the 
maximum  space  current  which  can  be  obtained  for  a  given  diff- 
erence of  potential  between  filament  and  plate  depends  upon  the 
shape,  dimensions  and  spacing  of  the  electrodes.  For  a  tube 
having  a  cylindrical  plate  of  radius  r,  and  a  straight  filament 


224  ELECTRICITY  AND  MAGNETISM 

placed  along  its  axis,  the  current  per  unit  length  of  filament  is 
given  by  the  expression1 

.  _  2\/2 
9 

where  V  is  the  difference  of  potential  between  filament  and  plate 

/> 

and  -  is  the  ratio  of  the  charge  to  the  mass  of  the  electron. 

/? 

Substituting  numerical  values  for  — >  and  expressing  V,  i,  and  r  in 

volts,  amperes  and  centimeters  this  becomes 

VH 

i  =  14.65  X  10~6-  (3) 

The  two  element  tube  finds  its  chief  application  as  a  rectifier  and 
is  often  called  the  "Kenotron."  Current  can  flow  only  when  the 
plate  is  positive  with  respect  to  the  filament.  When  the  plate  is 
negative,  filament  electrons  are  driven  back  into  it  as  fast  as 
they  are  emitted,  and  so  long  as  the  plate  is  cold,  none  are  emitted 
there.  Consequently  there  can  be  no  reverse  current.  The  tube 
is  thus  a  perfect  rectifier  for  any  voltage  within  the  limits  of  the 
mechanical  and  dielectric  strength  of  the  parts  of  which  it  is  made. 
It  ceases  to  function  as  a  rectifier  however,  if  the  plate  becomes 
too  hot  for  it  also  then  becomes  a  source  of  electrons.  Even  at 
relatively  low  voltages,  electrons  acquire  velocities  of  many  thou- 
sand miles  per  second  in  passing  from  filament  to  plate  and  thus 
strike  it  possessing  very  appreciable  amounts  of  kinetic  energy 
which  is  converted  into  heat  by  bombardment.  In  fact  this 
effect  is  made  use  of  in  heating  the  plates  to  "outgas"  them 
during  the  evacuation  process. 

173.  Experiment  35.  Characteristics  of  the  Two  Element 
Electron  Tube. — Connect  the  apparatus  as  shown  in  Fig.  116, 
using  for  B  a  battery  of  flash  light  cells  or  a  motor  generator  set 
giving  an  E.M.F.  of  about  500  volts.  The  purpose  of  this 
experiment  is  to  obtain  the  two  sets  of  characteristic  curves  for 
the  Kenetron  rectifier  illustrated  in  Figs.  117  and  118.  Ascertain 
from  the  instructor  the  normal  filament  current  for  the  tube,  and 
using  this  and  two  smaller  ones  take  plate  potential-space  current 
characteristics  for  each,  varying  plate  potentials  from  0  to  500 
volts.  Determine  also  the  filament  current — space  current 
characteristics  for  three  different  values  of  plate  potential. 

1  LANGMUIR,  Gen.  Elec.  Rev.,  1915,  p.  330. 


ELECTRON  TUBES  225 

Report. — Plot  the  two  sets  of  curves  as  indicated.  Explain 
what  is  meant  by  voltage  saturation  and  space  charge.  From 
formula  3  compute  the  dimensions  of  a  tube  that  would  carry  one 
quarter  of  an  ampere  with  a  difference  of  potential  of  500  volts. 

174.  The  Three -element  Electron  Tube. — In  the  discussion  of 
the  two-element  tube  the  dependence  of  space  current  upon  fila- 
ment temperature  .and  plaife  potential  was  described,  and  it 
was  pointed  out  that  its  principal  application  is  in  the  rectification 
of  high  voltage  alternating  currents.  By  changing  the  tempera- 
ture of  the  filament,  thus  regulating  the  supply  of  available 
electrons  it  also  serves  as  a  means  of  controlling  currents.  In 
this  way,  it  acts  as  an  electrical  valve  which  may  be  opened  or 
closed  to  any  desired  fraction  of  its  current  carrying  capacity. 
Since,  however,  filament  temperatures  do  not  respond  immedi- 
ately to  changes  in  heating  current,  this  action  is  sluggish,  and 
it  can  not  be  used  in  this  way  to  produce  current  variations  that 
are  at  all  rapid. 

It  has  been  found  that  the  space  current  may  be  controlled 
with  remarkable  ease  by  the  introduction  between  the  filament 
and  plate  of  a  third  electrode  in  the  form  of  a  grid  or  mesh  of  fine 
wires  through  which  the  electrons  must  pass  on  their  way  from 
filament  to  plate.  Such  an  arrangement  is  shown  in  Fig.  119. 
If  a  difference  of  potential  is  established  between  the  filament  and 
grid  by  means  of  the  battery  C,  the  grid  tends  to  accelerate  or 
retard  the  electrons  of  the  space  current  according  as  it  is  positive 
or  negative  with  respect  to  the  filament.  It  thus  counteracts  or 
increases  the  effect  of  the  space  charge.  The  operation  of  the 
three-element  tube  may  be  best  described  by  means  of  the  curve 
of  Fig.  120,  which  shows  the  relation  between  the  plate  current 
and  the  grid  volts  and  is  known  as  the  "  static  characteristic." 
If  the  grid  is  disconnected  from  the  circuit,  the  tube  behaves  as 
the  ordinary  two-element  device  in  which  the  space  current  is 
limited  either  by  electron  emission  of  the  filament  or  by  space 
charge.  Assuming  there  is  available  a  sufficient  supply  of  elec- 
trons so  that  the  space  charge  is  the  controlling  factor,  a  negative 
potential  placed  upon  the  grid  adds  to  the  retarding  action  of  the 
space  charge,  and  the  plate  current  is  reduced,  and  may  even 
be  made  zero,  if  the  grid  is  sufficiently  negative.  Again,  if  the 
grid  is  positive,  it  neutralizes  to  a  certain  extent  the  effect  of  the 
space  charge,  causing  an  increase  of  the  space  current.  The  space 
current  can  not  continue  increasing  indefinitely,  for  even  though 

15 


226  ELECTRICITY  AND  MAGNETISM 

the  space  charge  were  completely  neutralized  by  positive  charges 
on  the  grid,  the  current  would  be  limited  by  the  electron  supply  at 
the  filament.  This  accounts  for  the  horizontal  part  of  the  static 
characteristic.  If  a  higher  voltage  is  applied  to  the  plate,  the 
characteristic  curve  is  not  changed  in  shape,  but  is  shifted  toward 
the  left.  This  is  because  larger  negative  grid  voltages  are  required 
to  reduce  the  space  current  to  a  given  value. 

This  method  of  controlling  the  space  current  has  a  number  of 
advantageous  features.  In  the  first  place,  it  requires  the  expendi- 
ture of  exceedingly  small  amounts  of  energy.  If  the  grid  is  nega- 
tive with  respect  to  the  filament,  no  electrons  strike  it  and 
consequently  no  current  flows  through  the  battery  C,  hence  the 
only  energy  drawn  from  it  is  that  required  to  charge  the  con- 
denser formed  by  the  grid  and  filament,  which  is  negligible  in 
most  cases.  If,  however,  the  grid  is  positive  with  respect  to  the 
filament,  a  few  electrons  strike  it  and  a  current  is  drawn  from  C 
which  then  supplies  energy  to  the  tube.  If,  however,  the  grid 
wires  are  very  fine,  this  current  may  be  made  quite  small  even 
though  relatively  large  positive  potentials  are  impressed  on  the 
grid.  The  battery  B  may  be  one  of  high  voltage  and  the  space 
current  will  therefore  have  large  amounts  of  power  associated 
with  it.  Accordingly,  by  the  expenditure  of  small  amounts  of 
power  in  the  grid  circuit,  large  amounts  of  power  in  the  plate 
circuit  may  be  controlled,  and  the  device  constitutes  a  relay 
having  a  large  energy  ratio. 

In  the  second  place,  the  response  of  the  plate  current  to  changes 
in  grid  potential  is  exceedingly  quick,  almost  instantaneous.  If 
the  time  required  for  an  electron  to  travel  from  the  filament  to 
the  plate  is  computed  by  eq.  (3)  of  chap.  XIII,  it  is  found  that  for 
an  ordinary  tube  with  moderate  plate  voltages  it  is  of  the  order 
of  one  hundredth  of  a  millionth  of  one  second.  This  then  is  the 
order  of  the  time  lag  to  be  expected.  For  this  reason  it  may  be 
regarded  as  a  relay  with  no  moving  mechanical  parts  and  is 
therefore  without  inertia  in  its  action. 

Again,  there  exists  for  a  considerable  range,  a  linear  relation 
between  grid  potential  and  plate  current  so  that  the  variations  in 
plate  current  are  faithful  reproductions  of  the  changes  in  grid 
potential  and  thus  the  device  is  a  distortionless  amplifier. 

175.  Experiment  36.  Static  Characteristics  of  a  Three-element 
Electron  Tube. — Connect  the  apparatus  as  shown  in  Fig.  119. 
Use  for  the  filament  battery  A,  a  set  of  storage  cells,  furnishing 


ELECTRON  TUBES 


227 


from  10  to  20  volts  depending  upon  the  size  of  the  tube  to  be 
tested.  Ascertain  from  the  instructor  the  normal  heating  cur- 
rent for  the  filament,  and  be  careful  that  this  is  not  exceeded  at 
any  time  during  the  test.  If  the  filament  is  of  the  oxide  coated 
type,  it  should  be  operated  at  a  dull  red  heat,  but  if  it  is  a  tung- 
sten wire,  bring  it  up  to  about  the  same  brightness  as  the  ordinary 
vacuum  incandescent  lamp.^  B  may  be  a  battery  of  flash  light 
cells  giving  500  volts  or  a  motor  generator  set.  For  C  use  a 
battery  of  flash  light  cells  giving  about  60  volts.  Bring  the 
filament  up  to  normal  temperature,  and  apply  a  plate  voltage 
of  ^<j  normal.  Apply  a  sufficient  negative  voltage  to  the  grid 


Fio.  119. — Three  element  electron  tube. 

to  reduce  the  plate  current  approximately  to  zero.  Raise  the 
grid  volts  by  steps  to  zero  and  positive  values  and  note  the 
grid  and  plate  currents  for  each  setting.  Repeat  for  several 
values  of  plate  voltage  up  to  and  including  normal. 

Report. — Describe  the  three  element  electron  tube  and  outline 
its  principal  operation  features.  Plot  the  static  characteristic  for 
the  plate  voltages  studied,  also  the  grid  current  as  a  function  of 
grid  volts.  Sometimes  a  negative  grid  current  is  obtained. 
How  can  this  be  explained? 

176.  Amplification  Factor. — The  fact  that  the  three-element 
tube  may  be  used  as  a  relay  has  been  referred  to  several  times, 
and  it  is  necessary  to  define  accurately  what  is  meant  by  this 
statement.  By  a  relay,  is  meant  any  device  by  which  a  small 
amount  of  energy  may  be  used  to  turn  on  and  off  or  control  a 
much  larger  source  of  energy.  In  the  case  of  the  electron  tube, 
the  source  of  energy  is  the  plate  battery  and  the  grid  is  the  gate 
by  which  it  is  controlled.  Considering  now  the  plate  and  grid 


228 


ELECTRICITY  AND  MAGNETISM 


circuits,  it  is  obvious  that  we  may  be  interested  in  the  relative 
values  of  either  the  power,  the  currents,  or  the  voltages  existing 
in  these  circuits,  and  that  we  may  accordingly  refer  to  either  the 
power  amplification,  the  current  amplification,  or  the  voltage 
amplification.  The  meaning  of  the  first  two  of  these  expressions 
is  obvious;  for  example,  by  power  amplification  is  meant  the 
ratio  of  the  change  in  power  drawn  from  the  plate  battery  to 
the  change  in  power  supplied  to  the  grid,  and  a  corresponding 
meaning  is  given  to  current  amplification. 

However,  in  the  ordinary  use  of  the  tube,  the  voltage  of  the 
plate  battery  remains  constant,  and  the  meaning  of  the  voltage 
amplification  factor  is  not  so  evident.  The  significance  of  this 

term  can  perhaps  be  un- 
derstood by  reference  to  a 
series  of  static  characteris- 
tics as  represented  in  Fig. 
121,  where  the  dependence 
of  space  current  upon  grid 
volts  for  a  series  of  plate 
potentials,  at  50  volts  in- 
tervals, is  shown.  Sup- 
pose, for  example,  the  plate 
voltage  is  100  and  the  grid 
volts  zero.  The  space 
current  is  then  10  milli- 
amperes.  It  is  desired  to 
increase  the  space  current 

to  20  milliamperes.  This  may  be  done  either  by  raising  the  plate 
voltage  to  150  or  the  grid  voltage  to  5.  Thus  an  increase  of  5 
volts  on  the  grid  produces  the  same  change  in  the  space  current 
as  an  increase  of  50  volts  on  the  plate.  The  voltage  amplification 
factor  in  this  case  is  said  to  be  10,  since  one  volt  on  the  grid  is 
equivalent  to  10  volts  on  the  plate. 

A  working  equation  connecting  these  quantities  may  be 
deduced  as  follows.  It  was  shown  in  eq.  (2)  that  for  the  two- 
element  tube,  the  plate  current  is  proportional  to  the  % 
power  of  the  plate  voltage,  i.e.,  Ip  =  aV^,  where  a  is  a  constant. 
Since  a  change  in  grid  voltage  is  more  effective  by  a  certain 
factor,  which  we  will  call  k,  in  producing  a  change  in  plate  current 
than  a  change  in  the  plate  voltage,  it  follows  that  the  plate  cur- 
rent in  a  given  tube  on  which  there  is  acting  a  plate  voltage  Ep 


Grid  Volts 

FIG.  120. — Characteristic  for  three  element 
electron  tube. 


ELECTRON  TUBES 


229 


and  a  grid  voltage  Eg  is  just  the  same  as  though  it  were  a  two- 
element  tube  with  a  plate  voltage  Ep  +  kEg.  The  expression 
for  the  current  then  becomes 

IP  =  a(Ep  +  kEa)*  (4) 

Since  eq.  (2)  refers  to  the  case  in  which  there  is  an  abundance  of 
electrons  at  the  filament  and  the  current  is  limited  only  by  the 
space  charge,  eq.  -(4)  holds'5  only  for  the  left-hand  part  of  the 
characteristic,  i.e.,  up  to  the  bend. 

Referring  to  Fig.  121,  it  is  seen  that  the  static  characteristics 
all  have  a  point  of  inflection,  and  that  for  a  considerable  portion 
each  side  of  this  point,  the  curve  is  nearly  a  straight  line.  If  the 


—          25       20       10       5        0         5        10       15      20       23     + 

Grid  Volts 
FIG.  121. — Dependence  of  static  characteristics  upon  plate  potential. 

tube  is  used  as  a  distortionless  amplifier,  it  is  necessary  that 
the  range  of  applied  grid  volts  should  not  appreciably  exceed  the 
linear  part.  In  this  case,  the  simplified  equation 

Ip  =  a(Ep  +  kEa)  (5) 

may  be  used  in  which  a  is  the  filament  to  plate  conductance  of  the 
tube,  and  is  the  slope  of  the  linear  part  of  the  characteristic. 
If  Eg  is  sufficiently  negative,  the  plate  current  is  zero.  Calling 
this  value  Ego  we  have 


k  =  - 


E, 


(6) 


It  is  obvious  that  the  amplification  for  a  given  tube  depends 
upon  the  spacing  of  the  grid  wires.     If  these  wires  are  far  apart, 


230  ELECTRICITY  AND  MAGNETISM 

a  definite  change  of  voltage  is  not  as  effective  in  controlling  the 
electron  flow  as  though  the  meshes  were  smaller.  As  a  matter  of 
fact,  the  amplification  factor  is  inversely  proportional  to  the  dis- 
tance between  grid  wires.  Again,  if  the  grid  is  close  to  the  fila- 
ment so  that  it  acts  upon  the  electrons  before  they  have  gained 
appreciable  speeds,  it  is  more  effective  than  if  it  is  near  the  plate. 
Thus,  if  it  is  desired  to  construct  a  tube  with  a  large  voltage 
amplification  factor  it  should  have  a  grid  with  a  fine  mesh 
mounted  close  to  the  filament.  Tubes  having  amplification 
factors  as  large  as  100  have  been  constructed,  but  in  actual  prac- 
tice factors  from  10  to  20  are  more  common. 

A  simple  method  for  obtaining  the  amplification  factor  of  a 
tube  is  to  impress  upon  the  plate  a  certain  positive  potential  and 
then  apply  to  the  grid  a  negative  potential  sufficient  to  reduce  the 
plate  current  to  zero.  The  ratio  of  the  plate  and  grid  potentials 
is  then  the  amplification  factor  of  the  tube  for  this  particular 
plate  voltage.  It  is  found  in  practice  that  the  amplification 
factor  of  a  tube  is  not  constant  but  varies  with  the  plate  and  grid 
potentials  used.  This  is  due  to  the  fact  that  the  average  distribu- 
tion of  electrons  between  the  plate  and  filament  changes  with  the 
potentials  on  the  grid  and  plate  which  in  effect,  changes  their 
relative  positions.  By  taking,  in  this  manner,  measurements 
over  a  series  of  values  of  plate  voltage  a  fair  idea  of  the  behavior 
of  the  tube  may  be  obtained. 

While  the  method  just  described  yields  results  sufficiently 
accurate  for  many  purposes,  it  has  nevertheless  one  serious  error. 
Unless  the  tube  is  very  carefully  designed,  it  does  not  have  a 
sharp  "cut  off."  That  is,  the  characteristic  curve  does  not  pro- 
ceed straight  down  to  the  axis,  but  slopes  off  and  approaches  it 
gradually.  The  actual  negative  grid  potentials  required  to 
reduce  the  plate  current  to  zero  are  much  larger  than  would  be 
obtained  by  continuing  the  straight  portion  of  the  characteristic 
until  it  intercepts  the  horizontal  axis.  In  actual  use,  this  inter- 
cept value  is  the  one  which  is  effective.  A  dynamic  method  in 
which  this  error  is  eliminated  has  been  devised  by  Miller.1 
His  circuit  is  shown  in  Fig.  122.  The  tube  is  connected  in  the 
ordinary  way  with  a  telephone  receiver  in  the  plate  circuit, 
and  potentials  supplied  to  the  plate  and  grid  by  the  batteries  B 
and  C  respectively.  By  properly  adjusting  the  values  of  these 
voltages,  the  tube  may  be  set  at  any  point  on  the  characteristic 

1  J.  H.  MILLER,  Proc.  Inst.  of  Radio  Engineers,  vol.  12,  1918,  p.  171. 


ELECTRON  TUBES 


231 


curve.  Included  in  the  grid  and  plate  circuits  are  the  resistances 
Ri  and  Ri  across  which  is  connected  the  secondary  of  a  telephone 
transformer  T.  When  an  alternating  current  is  supplied  to  the 
primary  of  this  transformer,  small  alternating  voltages,  i.e.,  the 
resistance  drops  across  RI  and  R*,  are  introduced  into  the  grid 
and  plate  circuits  respectively.  It  is  obvious  from  the  con- 
nections that  when^the  addi&onal  voltage  on  the  plate  is  positive 
that  on  the  grid  is  negative  and  vice  versa.  By  changing  the 
relative  values  of  R  i  and  Rz  the  ratio  of  these  voltages  may  be 


I 1 


FIG.  122. — Dynamic  method  for  amplification  factor. 

made  to  have  any  desired  value.  If  it  is  such  that  the  added 
grid  potential  just  balances  that  added  to  the  plate,  there  will 
be  no  change  in  the  steady  plate  current  and  consequently  no 
sound  in  the  phones.  The  amplification  factor  k  is  then  the 
ratio  of  R%  to  R\.  The  advantage  of  this  method  is  that  it  mea- 
sures the  amplification  factor  while  the  tube  is  operating  in  the 
same  manner  as  when  actually  used  in  practice.  The  dependence 
of  the  amplification  factor  upon  the  plate  and  grid  volts  may  thus 
be  easily  and  quickly  obtained. 

177.  Experiment  37.  Amplification  Factor  of  a  Three-element 
Electron  Tube. — Connect  the  apparatus  as  shown  in  Fig.  122,  using 
for  P  a  pair  of  high  resistance  head  receivers.  The  source  B 
should  furnish  a  voltage  equal  to  the  maximum  for  which  the  tube 


232  ELECTRICITY  AND  MAGNETISM 

is  designed,  and  if  a  power  tube  is  under  test,  may  be  a  high 
voltage  generator.  C  should  consist  of  a  battery  of  small  flash 
light  cells.  The  A.  C.  supply  should  have  a  frequency  high 
enough  to  give  a  good  clear  note  in  the  phones,  and  the  voltages 
across  Ri  and  R2  should  be  low  enough  so  that  the  operating 
point  moves  only  a  small  amount  along  the  static  characteristic 
curve.  Make  two  tests.  First  hold  the  grid  volts  at  some 
predetermined  value,  and  measure  the  amplification  factor  for  a 
series  of  plate  voltages  ranging  from  a  small  value  up  to  the 
maximum  for  which  the  tube  is  designed.  Next  hold  plate 
volts  at  normal  value  and  measure  the  amplification  factor  for  a 
series  of  values  of  grid  volts.  Check  the  results  of  the  first  series 
by  the  static  method  explained  above.  That  is,  for  each  different 
plate  voltage,  find  the  negative  grid  potential  required  to  reduce 
the  plate  current  to  zero. 

Report. — Plot  curves  showing  the  dependence  of  the  amplifica- 
tion factor  upon  both  the  plate  and  grid  potentials  by  the 
dynamic  method,  and  upon  plate  potentials  for  the  static 
method.  How  do  you  account  for  the  differences  between  these 
curves? 

178.  Internal  Plate  Resistance  of  a  Three-element  Electron 
Tube. — Following  the  amplification  factor,  the  next  most 
important  characteristic  of  an  electron  tube  from  the  standpoint 
of  operation  is  perhaps  its  internal  impedance.  It  is  a  well  known 
principle  of  electrical  practice  that  the  impedance  of  a  device 
should  equal  that  of  the  circuit  on  which  it  operates.  Accord- 
ingly, in  designing  a  tube  to  operate  on  a  particular  circuit  or 
conversely  in  adjusting  a  circuit  to  fit  the  tube  which  is  supplying 
power  to  it,  it  is  necessary  to  know  the  plate  to  filament  impe- 
dance of  the  tube.  The  mechanism  by  which  the  vacuous  space 
offers  resistance  may  be  understood  by  the  following  considera- 
tion. When  a  current  flows  through  a  conductor,  heat  is 
developed  within  it.  This  energy  is  furnished  by  the  driving 
electric  field  which  urges  the  electrons  along  through  the  con- 
ductor. Resistance,  in  this  case,  is  due  to  a  direct  interference 
with  the  motion  of  electrons.  As  a  consequence  of  this  view  of 
the  nature  of  resistance,  it  might  at  first  be  thought  that  a  perfect 
vacuum  would  be  a  perfect  conductor  of  electricity  since  there  is 
nothing  to  interfere  with  the  free  motion  of  electrons.  That  this 
however,  is  not  the  case  is  at  once  evident  when  one  remembers 
that  relatively  large  voltages  are  necessary  to  cause  small  currents 


ELECTRON  TUBES  233 

to  flow  through  the  ordinary  electron  tubes,  even  when  the  condi- 
tions are  far  removed  from  those  of  current  saturation.  More- 
over, the  fact  that  it  is  easy  to  heat  the  plate  red  hot  by  the  passage 
of  current,  indicates  that  it  is  accompanied  by  a  consumption  of 
energy. 

When  an  electron  is  emitted  by  the  heated  filament,  it  finds 
itself  in  the  electrostatic  field  existing  between  filament  and 
plate,  and  it  is  at  once  accelerated  toward  the  plate.  Since  the 
electron  possesses  mass,  it  necessarily  gains  kinetic  energy  as  it 
moves  toward  the  plate.  This  energy  is  abstracted  from  the 
electric  field  which  accelerates  it.  When  the  electron  strikes  the 
plate,  it  possesses  a  velocity  of  the  order  of  several  thousand  miles 
per  second  even  under  moderate  potential  differences  At  the 
plate  it  is  suddenly  brought  to  rest  and  its  kinetic  energy  of 
motion  is  converted  into  heat  energy  of  the  molecules  of  the 
plate.  While  the  tube  does  not  possess  resistance  in  quite  the 
same  way  that  an  ordinary  metallic  conductor  does,  it,  never- 
theless, consumes  energy  when  a  current  passes,  and  it  is  cus- 
tomary to  speak  of  its  resistance  and  to  define  it  on  the  basis  of 
the  energy  it  consumes.  Thus,  if  /  is  the  current  flowing  through 
the  tube,  and  W  the  watts  consumed  by  it,  its  resistance  R  is 
defined  to  be  such  that 

W  =  PR  (7) 

Since  this  is  the  same  equation  as  holds  for  the  power  converted 
into  heat  by  the  ordinary  conductor,  we  may  determine  the 
resistance  of  the  tube  by  the  voltage  required  to  furnish  a  given 
current  through  it.  An  application  of  Ohm's  law  to  correspond- 
ing values  of  plate  volts  and  plate  current  as  read  from  the  static 
characteristics  shows  that  the  resistance  of  a  tube  is  not  constant 
but  depends  upon  the  values  of  both  the  plate  and  grid  potentials, 
and  also  upon  the  electron  emission  from  the  filament  in  case 
saturation  voltages  are  used.  It  is  necessary  therefore  to  define 
the  resistance  of  the  tube  for  a  particular  point  in  the  char- 
acteristic curve.  This  is  done  by  saying  that  the  resistance  of  the 
tube  is  the  ratio  of  the  change  in  plate  volts  to  the  change  in  plate 
current  produced  by  it,  when  this  change  is  made  vanishingly 
small.  That  is 


Thus  the  resistance  is  the  reciprocal  of  the  slope  of  the  plate 
potential,  plate  current  characteristic.     Since  this  curve  is  seldom 


234 


ELECTRICITY  AND  MAGNETISM 


taken  in  practice,  R  may  be  obtained  from  the  plate  current-grid 
potential  characteristic  by  remembering  that 

Ep  =  kE,  (9) 


whence 
Therefore 


=  kdEg 
dEg 


R  =  k 


(10) 
(11) 


The  internal  plate  resistance  is  then  the  product  of  the  amplifica- 
tion factor  and  the  reciprocal  of  the  slope  of  the  plate  current-grid 
potential  characteristic. 

While  this  method  is  satisfactory  for  many  purposes,  it  is  open 
to  the  objection  that  it  requires  a  determination  of  the  amplifica- 
tion factor  k.  A  dynamic  null  method  has  been  employed 
by  Ballantine1  in  which  the  resistances  may  be  measured 


I — wwW     -=- 


AAAAAA 


A.C.Supply 
FIG.   123. — Connections  for  measuring  resistance  of  tube. 

directly.  The  connections  for  this  circuit  are  shown  in  Fig.  123. 
It  will  be  noted  that  the  arrangement  is  essentially  a  Wheatstone 
bridge  in  which  the  plate  to  filament  path  through  the  tube  is  one 
of  the  arms.  Because  of  the  battery  B  a  steady  current  flows 
through  all  four  arms  of  the  bridge  and  also  through  the  phones. 
The  phones,  however,  respond  only  to  the  variable  currents 

1  BALLANTINE,  Proc.  Inst.  Radio  Engineers,  vol.  7,  1919,  p.  129. 


ELECTRON  TUBES  235 

furnished  by  the  A.C.  supply.     The  resistance  thus  measured  will 
be  those  defined  by  eq.  (8). 

179.  Experiment  38.    Plate-filament  Resistance  of  an  Electron 
Tube. — Connect  the  apparatus  as  shown  in  Fig.  123.     As  a  source 
of  alternating  voltage  use  any  oscillator  giving  a  good  clear  note 
furnishing  an  E.M.F.   of*  about   10  volts.     The  resistance  R3 
should  be  of  the  same  order  ftf  magnitude  as  the  tube  under  test, 
i.e.,  several  thousand  ohms.     Ascertain  the  normal  plate  voltage 
for  the  tube,  and  make  a  series  of  measurements  of  internal 
resistance  varying  the  grid  volts  over  a  considerable  range,  both 
positive  and  negative.     Repeat  using  plate  voltages  three-quar- 
ters,   one-half   and   one-quarter  normal.     Disconnect  the  tube 
from  the  bridge  and  determine  its  static  characteristic  for  normal 
plate  voltage. 

Report. — Plot  internal  resistance  as  a  function  of  grid  volts  for 
the  four  series  of  observations.  Plot  the  static  characteristic 
and  check  your  results  by  the  first  method  described.  The 
amplification  factor  may  be  obtained  by  extending  the  straight 
portion  of  the  characteristic  and  taking  its  intercept  on  the 
horizontal  axis  as  the  value  of  the  grid  volts  necessary  to  reduce 
the  plate  current  to  zero.  See  Art.  176. 

180.  The  Tungar  Rectifier. — In  the  case  of  tubes  operated 
on  a  pure  electron  discharge,  it  is  possible,  at  best,  to  obtain 
currents  of  but  a  fraction  of  an  ampere,  and  these  only  by  the 
employment  of  several  hundred  volts.     While  such  tubes  are 
satisfactory  for  the  rectification  of  high  voltage  currents  they 
are,  nevertheless,  unsuitable  for  cases  in  which  several  amperes  at 
low   voltage   are   required,    as,   for   example,    charging   storage 
batteries  from  ordinary  city  lighting  circuits.     For  this  purpose, 
a  satisfactory  tube,  known  as  the  tungar  rectifier    has    been 
developed   by  the  General  Electric  Co.1     It  is  a  two-element 
tube,  the  cathode  of  which  is  a  heated  tungsten  filament  in  the 
form  of  a  helix,  while  the  anode  is  a  conical  piece  of  tungsten 
mounted  about  3  mm.  from  the  filament.     Instead  of  a  vacuum, 
the  tube  contains  pure  argon  at  a  pressure  of  8  or  10  cms.  of 
mercury. 

The  purpose  of  the  argon  is  to  furnish  positive  ions  which 
neutralize  the  space  charge  encountered  in  pure  electron  tubes, 
and  thus  to  reduce  by  many  fold  the  voltage  required  to  maintain 
the  current.  Furthermore  the  positive  ions  take  part  in  trans- 

1  Gen.  Elec.  Rev.,  vol.  19,  No.  4,  1916,  p.  197. 


236  ELECTRICITY  AND  MAGNETISM 

porting  electricity  between  the  electrodes  and  thus  materially 
increase  the  carrying  capacity  of  the  tube.  In  the  early  attempts 
to  utilize  positive  ions,  it  was  found  that  many  gases  have 
injurious  effects.  For  example,  in  the  presence  of  oxygen,  the 
electron  emission  of  tungsten  is  cut  down  to  a  small  fraction  of 
what  it  is  in  high  vacuum.  Again,  many  gases  unite  with  the 
heated  filament  forming  compounds,  which  are  highly  volatile 
at  normal  operating  temperatures  and  thus  cause  it  to  disinteg- 
rate. Furthermore,  when  a  gas  is  present  in  only  small  amounts, 
the  mean  free  path  of  the  positive  ions  may  be  so  great  that  they 
acquire  velocities  sufficient  to  chip  off  particles  of  the  filament 
softened  by  heating,  and  thus  hasten  its  disintegration.  By  use 
of  an  inert  gas  such  as  argon,  the  first  two  difficulties  are  over- 
come and  by  shortening  the  mean  free  path  by  using  relatively 
high  pressures,  the  speeds  are  so  reduced  by  frequent  collisions 
that  the  disintegration  by  bombardment  is  insignificant. 

In  order  to  avoid  the  formation  of  volatile  compounds  it  is 
necessary  that  the  argon  be  very  pure,  and  in  the  early  tubes 
great  pains  were  taken  to  secure  this.  It  has  been  found  possible 
to  mount  within  the  tube,  usually  on  one  of  the  filament  leads, 
substances  which  react  chemically  with  the  impurities,  which 
thus  keep  the  argon  in  a  pure  state.  For  the  larger  sized  tubes, 
a  graphite  anode  mounted  on  a  tungsten  support  is  often 
used,  and  the  purifying  agent  may  then  be  introduced  in  the 
anode.  As  impurities  are  given  off  from  the  electrodes  or  interior 
walls,  the  drop  across  the  arc  increases,  liberating  more  heat  at 
the  anode,  which  thus  causes  vapors  to  be  given  off  by  the  purify- 
ing agent  and  in  this  way  the  argon  is  maintained  in  a  state  of 
high  purity. 

After  the  arc  has  once  been  started,  the  filament  may  be  kept 
heated  by  positive  ion  bombardment  after  the  heating  current 
has  been  shut  off.  In  this  case,  the  arc  confines  itself  to  a  very 
limited  portion  of  the  filament.  This  spot  wastes  away  more 
rapidly  than  the  rest  of  the  filament  and  the  life  of  the  tube  is 
materially  shortened  when  operated  in  this  way.  For  the  larger 
sized  tubes,  i.e.,  those  with  a  current  capacity  of  20  to  40  amperes, 
a  fine  tungsten  point  is  independently  mounted  close  to  the  fila- 
ment. This  may  be  heated  to  a  high  temperature  by  using  it 
as  anode  with  the  filament  as  cathode.  If  the  connections  are 
then  shifted,  this  hot  point  may  be  used  as  the  cathode  against 
the  regular  anode,  its  temperature  being  maintained  by  positive 


ELECTRON  TUBES 


237 


ion  bombardment  as  just  explained.  The  filament  serves  then 
as  a  starting  device  only  and  the  tube  has  an  exceedingly  long 
life.  Since  relatively  large  amounts  of  power  are  consumed  by 
the  filament  current,  it  might  be  expected  that  the  latter  method 
of  operation  would  result  tirf  a  material  increase  in  efficiency. 
This  is  not  the  case,  since  thg  voltage  across  the  arc  rises  when 
the  filament  current  fs  cut  off,  and  the  resulting  increase  in  energy 
consumption  in  the  arc  itself  practically  balances  the  saving 
effected  in  the  filament.  Commercial  sets  are  usually  made  for 
the  purpose  of  charging  automobile  storage  batteries  with  a 
maximum  E.M.F.  of  60  volts  directly  from  110  volt  alternating 
current  circuits.  To  avoid  losses  in  controlling  rheostats,  a  step 
down  transformer  is  mounted  within  the  case  to  reduce  the  A.C. 
voltage  to  the  desired  value  before  rectifying  it.  A  separate  low 
voltage  winding  is  included  for  heating  the  filament. 

181.  Experiment    39.     Study    of   the    Tungar    Rectifier. — For 
simplicity  of  operation,  obtain  the  characteristic  curves  by  the 


FIG.  124. — Connections  for  tungar  rectifier. 

use  of  direct  currents.  Mount  the  tube  in  a  special  socket  and 
connect  it  in  circuit  as  shown  in  Fig.  124.  Ascertain  the  normal 
heating  current  for  the  filament  and  be  careful  not  to  exceed  this 
value.  With  this  arrangement  four  curves  are  to  be  taken:  (a) 
The  volt-ampere  characteristic  for  the  arc ;  (6)  the  efficiency  of  the 
rectifier  with  external  filament  heating  current;  using  30  volts 
on  the  plate;  (c)  the  efficiency  of  the  rectifier  with  filament 
heated  by  positive  ion  bombardment,  30  volts  on  plate,  and  (d) 
same  as  (6)  using  60  volts  on  plate.  In  all  cases  vary  the  arc 


238 


ELECTRICITY  AND  MAGNETISM 


current  by  means  of  the  rheostat  R  through  as  wide  ranges  as  the 
arc  will  permit.  The  power  consumed  by  R  is  taken  as  the  load 
or  useful  output  of  the  device.  Next  place  the  tube  in  the  socket 
of  the  regular  rectifier  set  and  make  an  efficiency  run  using  the 
110  volt  A.  C.  circuit  as  a  source  of  power.  Measure  the  input 
by  means  of  a  wattmeter  and  the  output  by  the  volt-ampere 
product  for  the  load  rheostat  R.  Vary  the  load  amperes  through 


FIG.   125.- 


A.C. Supply 

-Connections  for  rectifier  mounted  in  commercial  set. 


as  wide  a  range  as  possible.  The  connections  for  this  test  are 
shown  in  Fig.  125.  Before  starting  the  test  open  up  the  housing 
for  the  set  and  study  carefully  the  internal  connections. 

Report. — Plot  the  volt-ampere  characteristic  for  the  tungar 
rectifier,  also  the  various  efficiency  curves  as  a  function  of  the 
load  current.  Is  there  any  similarity  between  the  volt-ampere 
characteristics  for  the  tungar  rectifier  and  that  of  the  ordinary 
carbon  arc. 


CHAPTER  XV 


PHOTOMETER1  ANB  OPTICAL  PYROMETER 

182.  Intensity  of  Radiation. — The  brightness  of  light,  as 
estimated  by  the  eye,  is  not  capable  of  precise  measurement, 
since  it  depends  to  a  large  extent  upon  the  color  of  the  light  and 
the  sensitiveness  of  the  eye  which  receives  it.  Accordingly,  the 
only  consistent  way  in  which  intensity  may  be  specified  is  in 
terms  of  energy.  Proceeding  on  this  basis,  the  intensity  of 
waves,  whether  they  are  those  of  sound,  light  or  of  any  other  type, 
is  measured  by  the  amount  of  energy  passing  per  second  through 
a  square  centimeter  of  area  at  right  angles  to  the  direction  of 
propagation.  If  there  is  no  loss  in  the  medium,  and  if  the 
medium  contributes  nothing  to  the  intensity,  the  same  quantity 
of  energy  will  persist  in  a  given  wave  no  matter  how  far  it  travels, 
or  how  the  dimensions  and  form  of  the  wave  front  may  change  as 
it  advances. 

The  variation  of  intensity  with  distance  from  the  source 
depends  upon  the  shape  of  the  wave  front,  or  what  amounts  to 
the  same  thing,  the  number  of  dimen- 
sions in  which  the  wave  spreads  out. 
For  example,  if  the  wave  front  is 
plane,  as  in  the  case  of  a  sound  wave 
travelling  along  a  speaking  tube,  or 
the  beam  from  a  searchlight,  the 
wave  front  maintains  a  constant 
area,  and  the  intensity  is  independent 
of  the  distance  from  the  source. 
Again,  if  a  pebble  is  dropped  in  the 
lake,  waves  travel  outward  in  circles 
and  are  propagated  in  two  dimen- 
sions. In  this  case  the  energy  remains  constant  in  a  circle  which 
increases  as  the  distance  from  the  center  and  the  intensity  varies 
inversely  as  the  distance  from  the  source.  In  the  case  of  spherical 

1  DUFF,  Text  Book  of  Physics,  arts.  259,  637-639,  724. 
KARAPETOFF,  Experimental  Electrical  Engineering,  arts.  205-211. 
NUTTING,  Outlines  of  Applied  Optics,  p.  169. 

239 


FIG.   126. — Propagation  of 
spherical  waves. 


240  ELECTRICITY  AND  MAGNETISM 

waves  with  which  we  are  particularly  concerned  here,  the  energy 
emitted  per  vibration  of  the  source  is  confined  within  a  spherical 
shell  whose  thickness  is  that  of  one  wave  length,  and  this  remains 
constant  as  the  wave  advances.  Let  0,  Fig.  126,  be  a  source 
from  which  waves  are  sent  out  in  all  directions.  Let  S  be  the 
strength  of  the  source,  i.e.,  the  amount  of  energy  emitted  per 
second.  Also  let  di  and  d2  be  the  radii  of  a  given  wave  at  two 
different  distances  from  the  source  and  let  /i  and  /2  be  the 
corresponding  intensities.  Then 

S  =  4Tdi271  =  47rd2272  (1) 

whence 

^  -  ^  (2) 

h  ~  di2 

Thus  the  intensity  varies  inversely  as  the  square  of  the  distance 
from  the  source. 

183.  The  Photometer. — An  instrument  for  the  comparison 
of  two  sources  of  light  is  called  a  photometer.  While  the  eye  is 
unable  to  estimate  absolute  intensities  at  all  accurately,  it  is, 
nevertheless,  quite  sensitive  to  differences  in  illumination. 


mimiiiiniii 


k- 


FIG.  127. — Principle  of  the  photometer. 

Accordingly,  if  light  from  two  different  sources  is  allowed  to  fall 
upon  a  screen  in  such  a  way  that  the  areas  of  the  separate  illumi- 
nations are  adjacent,  equality  in  the  two  intensities  may  be 
determined  by  the  disappearance  of  the  line  of  demarkation 
between  them.  An  instrument  for  this  purpose  may  be 
arranged  as  shown  in  Fig.  127  by  mounting  two  lamps  LI  and  L2, 
which  are  to  be  compared,  at  the  ends  of  a  bench  provided  with  a 
scale  along  which  runs  a  carriage  supporting  a  screen  of  white 
paper.  The  central  portion  of  this  screen  is  impregnated  with 
paraffine  which  renders  it  semitransparent.  This  spot  appears 
darker  than  its  surroundings  if  viewed  by  reflected  light,  but  it  is 
brighter  in  transmitted  light.  If,  however,  the  intensity  of 
illumination  is  the  same  on  both  sides,  the  spot  disappears  since 
the  amounts  transmitted  in  the  two  directions  are  equal. 


PHOTOMETER  AND  OPTICAL  PYROMETER 


241 


If  Si  and  $2  are  the  strengths  of  the  two  sources  and  d\  and  d2 
their  respective  distances  from  the  screen,  then  by  eq.  (1)  the 
ilumination  on  each  side  of  the  screen  is  given  by 


-         S,*-  d?  <4> 

If  one  of  the  sources,  e.g.,  S2  is  a  standard  lamp,  Si  may  be 
computed. 

184.  The  Lummer-Brodhun  Photometer. — A  comparator  con- 
siderably more  sensitive  than  the  grease  spot  screen  just  described 
has  been  developed  by  Lummer  and  Brodhun.  The  special 


FIG.   128. — Lummer-Brodhun  photometer. 

feature  of  this  instrument  is  the  optical  device  for  simultaneously 
viewing  the  two  sides  of  the  comparison  screen  W,  as  shown  in 
Fig.  128.  Light  from  each  side  is  reflected  by  two  mirrors  or 
prisms  M i  and  M2  so  as  to  enter  the  optical  system  AB.  This 
consists  of  two  totally  internally  reflecting  prisms  placed  back 
to  back.  The  reflecting  surface  of  one  is  plane,  while  that  of  the 
other  is  spherical  with  a  small  portion  ground  flat.  The  flat 
surface  of  the  latter  is  placed  in  optical  contact  with  the  former. 

16 


242  ELECTRICITY  AND  MAGNETISM 

Light  entering  either  of  these  prisms  and  striking  the  contact 
surface  will  be  transmitted,  but  light  striking  any  portion  of  the 
reflecting  surfaces  backed  by  air  will  be  totally  internally 
reflected.  Light  emerging  from  the  prism  B  consists  of  two  parts, 
that  from  the  contact  portion  of  the  two  prisms  and  that  from 
the  surrounding  area.  The  former  comes  entirely  from  the  left- 
hand  side  of  W  while  the  latter  is  from  the  right-hand  side.  If  a 
telescope  is  placed  at  T  and  focused  on  the  contact  area  of  the 
two  prisms,  the  central  portion  appears  brighter  or  darker  than 
the  surroundings  according  as  the  illumination  of  the  left-  or  the 
right-hand  side  of  W  is  more  intense,  but  the  entire  field  appears 
uniformly  illuminated  when  a  balance  is  secured. 

A  convenient  form  of  laboratory  instrument  is  one  in  which  a 
single  socket,  to  receive  in  succession  the  unknown  and  standard 
lamps,  is  mounted  at  a  fixed  distance  from  the  comparison  box. 
On  the  other  side  is  a  movable  socket  containing  a  small  six  volt 
lamp  for  comparison  purposes.  The  distance  of  this  lamp  from 
the  screen  is  read  by  an  index  registering  on  a  fixed  scale.  A 
slow  motion  device  is  also  provided.  The  process  consists  then 
in  placing  the  unknown  lamp  in  the  fixed  socket  and  obtaining 
a  balance  by  moving  the  comparison  lamp  to  or  from  the  screen 
until  the  line  of  demarkation  between  the  outer  and  central  posi- 
tions of  the  field  of  the  telescope  disappears.  The  lamp  is  then 
replaced  by  the  standard  and  a  balance  again  obtained.  The 
screen  should  be  reversed  and  readings  taken  in  each  position 
and  averaged  to  eliminate  differences  in  reflecting  power  of  the 
two  sides.  The  equation  for  computing  the  strength  of  the 
unknown  lamp  may  be  derived  as  follows: 

Let  S,  U,  and  C  be  the  candle  powers  of  the  standard,  the 
unknown,  and  comparison  lamps,  respectively;  \etds  anddM  be  the 
distances  of  the  comparison  lamp  from  the  screen  when  balanced 
against  the  standard  and  unknown,  and  let  D  be  the  fixed  dis- 
tance of  both  standard  and  unknown  from  the  screen.  Then,  for 
the  two  balances,  the  following  equations  hold : 
U  _  D2  S  D2 

C  '"  du*  C  "  ~ds* 

Dividing  one  equation  by  the  other,  we  have 

V  -  d'2  (5) 

S"rf? 

Care  must  be  taken  to  maintain  the  same  voltage  on  the  compari- 
son lamp  throughout  the  test. 


PHOTOMETER  AND  OPTICAL  PYROMETER  243 

185.  Experiment    40.     Study    of    Incandescent   Lamps. — The 
purpose  of  this  experiment  is  to  determine,  as  a  function  of  the 
voltage  upon  which  they  are  operated,  the  candlepower,  wattage 
consumption,   watts   per   candlepower,   and   resistance   of  four 
lamps  differing  as  widely  as.  possible  in  design.     Each  lamp, 
including   the   comparison  lamp,   should   be   provided   with   a 
voltmeter  and  a  control  rheostat.     An  ammeter  should  be  placed 
in  series  with  the  unknown.     Use  five  different  voltages  between 
90  and  130.     Do  not  operate  the  lamps  at  the  higher  voltages 
longer   than  is   necessary  for  making  the   observations.     The 
standard  lamp  should  be  operated  only  at  the  voltage  for  which  it 
is  rated.     Make  several  settings  for  each  observation  using  the 
screen  in  both  the  direct  and  reversed  positions. 

Report. — Describe  the  Lummer-Brodhun  photometer  and  plot 
the  four  curves  indicated  for  each  lamp.  Why  does  a  tungsten 
lamp  reach  full  brilliancy  more  quickly  after  closing  the  switch 
than  a  carbon?  Why  does  the  gas-filled  lamp  have  a  higher 
efficiency  than  a  vacuum  lamp? 

THE    OPTICAL    PYROMETER1 

186.  General  Principles. — It  is  a  matter  of  common  experience 
that  when  a  body  is  heated  to  a  high  temperature  it  emits  light 
and  also  that  the  intensity  of  this  emitted  radiation  varies 
rapidly  with  the  temperature  of  the  source.     For  example,  a  small 
change  in  the  voltage  across  an  incandescent  lamp  produces  a 
relatively  large  change  in  the  brightness  of  the  filament.     Meas- 
urements show  that  a  body  at  1,500°  C.  emits  more  than  one 
hundred  times  as  much  as  it  does  at  1,000°  C.,  and  if  the  tempera- 
ture is  raised  to  2,000°  C.,  the  radiation  is  increased  more  than  two 
thousand  fold.     This  fact  is  often  made  use  of  in  the  measure- 
ment of  temperatures,  and  pyrometers  operating  on  this  principle 
have  the  marked  advantage  that  it  is  not  necessary  to  heat  any 
part  of  the  measuring  apparatus  to  the  temperature  of  the  body 
being  studied.     This  is  particularly  important  for  work  above 
1,600°  C.,  for  there  is  no  substance  which  retains  its  temperature 
measuring  properties  uniform  when  subjected  to  such  extreme 
heats.     Again,  the  products  of  combustion  in  furnaces  contami- 
nate  any   pyrometric   material   introduced,    thus   necessitating 
frequent  recalibrations. 

1  LECHATELIER  and  BURGESS,  Measurement  of  High  Temperatures,  pp. 
237-243,  291-303,  325-327,  336-337. 

GRIFFITHS,  Methods  of  Measuring  Temperature,  pp.  113-118. 


244  ELECTRICITY  AND  MAGNETISM 

The  radiation  method  of  measuring  temperatures,  however,  is 
complicated  by  the  fact  that  incandescent  bodies  differ  materially 
as  regards  both  the  intensity  and  quality  of  the  light  which  they 
emit.  For  example,  the  radiation  from  iron  or  carbon  is  much 
greater  than  that  from  such  substances  as  magnesia  or  polished 
platinum  at  the  same  temperature.  If  a  pyrometer  were  cali- 
brated by  measuring  the  radiation  from  one  substance  and  then 
used  to  measure  the  temperature  of  another  possessing  different 
radiating  properties  large  errors  would  result  in  many  cases. 

This  difference  in  radiating  properties  has  led  to  the  use  of 
" black  bodies"  as  standard  radiators  and  absorbers.  A  black 
body  is  defined  as  one  which  absorbs  all  the  radiation  falling  upon 
it,  and  it  therefore  neither  reflects  nor  transmits  any  radiation. 
It  also  has  the  property,  when  heated,  of  emitting  radiation  whose 
intensity  is  a  function  of  temperature  only  and  depends  in  no 
way  upon  the  physical  constants  of  the  material  of  which  it  is 
made.  Further,  the  intensity  of  the  radiation  from  a  black  body 
at  a  given  temperature  is  greater  than  that  from  any  other  body 
at  the  same  temperature. 

187.  Black  Body  Furnace. — Experimentally,  a  black  body  is 
very  closely  approximated  by  a  hollow  opaque  inclosure  with  a 
small  opening.  If  the  internal  area  of  the  inclosure  is  large 
compared  to  the  opening,  radiation  falling  upon  it  enters  the 
inclosure  and  is  reflected  diffusely  back  and  forth  so  many  times, 
that  it  is  practically  all  absorbed  before  any  can  emerge.  Again, 
if  the  walls  are  heated  uniformly  to  any  temperature,  the 
radiation  emerging  from  the  opening  has  been  reflected  back  and 
forth  so  many  times  that  it  no  longer  has  properties  characteristic 
of  the  material  of  the  walls.  Such  a  body  is  at  the  same  time  a 
perfect  absorber  and  a  perfect  emitter.  The  radiation  from 
a  crack  or  other  small  opening  in  an  ordinary  furnace  is  nearly 
black  body  radiation,  so  also  is  that  from  the  inside  of  a  narrow 
wedge  formed  by  folding  a  thin  metallic  ribbon  into  a  very  flat  V. 

A  black  body,  satisfactory  for  experimental  purposes,  is  made 
by  winding  a  porcelain  tube  with  thin  platinum  foil  through 
which  a  heating  current  may  be  passed.  The  center  of  the  tube 
is  closed  by  a  porcelain  disk  and  between  this  and  the  end, 
through  which  observations  are  made,  is  arranged  a  series  of  dia- 
phragms, also  of  porcelain,  whose  apertures  increase  in  diameter 
successively  toward  the  end.  These  minimize  the  disturbing 
effects  of  air  currents  and  increase  the  number  of  internal  reflec- 


PHOTOMETER  AND  OPTICAL  PYROMETER  245 

tions  which  the  radiation  must  make  before  it  emerges.  To 
protect  the  internal  tube  from  external  disturbances  and  reduce 
the  heat  losses  to  a  minimum,  it  is  surrounded  by  another  tube 
upon  which  is  wound  a  second  heating  coil  of  some  alloy  such  as 
nichrome  or  therlo.  Outside,  of  this  is  a  series  of  several  addi- 
tional tubes  with  air  spaces' between  them,  the  outer  one  usually 
being  surrounded  by^ powdered  magnesia.  By  properly  adjust- 
ing the  heating  currents  through  the  two  coils,  any  desired 
temperature  up  to  1,600°  C.  may  be  maintained  with  a  high 
degree  of  constancy.  The  temperature  of  the  black  body  is  usually 
measured  by  a  platinum,  platinum-rhodium  thermocouple,  the 
junction  of  which  is  supported  by  two  small  holes  through  the 
central  disk,  with  the  insulated  leads  passing  out  through 
the  rear  of  the  furnace. 

188.  Distribution  of  Energy  in  the  Spectrum. — If  one  measures 
the  total  energy  emitted  by  a  black  body,  he  finds  that  it  increases 
rapidly  as  the  temperature  is  raised.  The  law  connecting  black 
body  radiation  with  temperature  was  first  stated  by  Stefan  and 
later  deduced  theoretically  by  Boltzmann.  It  is 

E  =  ST*  (6) 

where  E  is  the  total  energy  radiated,  T  the  absolute  temperature, 
and  S,  a  constant  which  is  approximately  5.6  X  10~5,  ergs  per 
square  centimeter  per  second.  Although  this  law  is  rigidly  true 
only  for  a  black  body  it  is  found  to  hold  approximately  for  most 
surfaces,  the  constant  S  being  different  for  each. 

If  the  radiation  from  a  black  body  is  separated  out  into  a 
spectrum  and  the  energy  associated  with  each  wave  length  is 
measured,  it  is  found  that  not  only  is  there  a  continuous  change 
in  the  amount  of  energy  as  we  go  from  one  wave  length  to  another, 
but  also  that  the  distribution  of  energy  among  the  wave  lengths 
changes  as  we  vary  the  temperature.  Figure  129  gives  the  dis- 
tribution of  energy  among  the  wave  lengths  for  a  series  of 
temperatures.  It  will  be  noted  that  as  the  temperature  is  raised, 
the  energy  in  each  wave  length  increases  but  not  in  the  same 
proportion.  Also  that  the  wave  length  containing  the  maximum 
energy  decreases  as  the  temperature  is  raised.  This  is  in  accord 
with  the  common  observation  that,  starting  with  low  tempera- 
tures, a  body  appears  at  first  dull  red,  then  yellowish  or  cherry 
red,  and  finally  becomes  " white  hot"  as  extreme  temperatures 
are  reached.  Wien  has  shown  that  the  wave  length  for  maximum 


246 


ELECTRICITY  AND  MAGNETISM 


energy  and  the  absolute  temperature  are  connected  by  the 
simple  law 

*  max?"    =    COnst.  (7) 

He  has  also  shown  that  the  distribution  of  the  energy  among  the 
wave  lengths  at  a  given  temperature,  as  illustrated  by  Fig.  129, 
follows  very  closely  the  law 


(8) 


140 


120 


100- 


°4 


I  . 

40- 
20- 


.005 


.006 


0  .001  .002  .003  .004 

Wave  Length  in  Millimeters 
FIG.  129.  —  Energy  distribution  for  a  black  body. 

where  EX  is  the  energy  in  the  wave  length  interval  X  to  X  +  d\; 
e  is  the  base  of  the  Naperian  logarithms;  T  the  absolute  tem- 
perature, and  Ci  and  c%  are  constants.  For  other  radiating 
surfaces,  it  is  found  that  E\  follows  very  closely  the  above  law 
but  different  constants  must  be  used. 

189.  Application  to  Pyrometry.  —  It  is  obvious  that  any  of  the 
three  equations  just  given  might  be  used  to  measure  tem- 
peratures. It  is  found,  however,  that  eq.  (8)  is  most  suitable,  and 
when  it  is  applied,  only  one  wave  length  is  used,  or  at  least  only 
those  lying  within  a  very  restricted  range.  This  equation  lends 
itself  more  easily  to  calculation  if  it  is  put  in  the  form  : 


log,,  fix  =  k  -       , 


(9) 


PHOTOMETER  AND  OPTICAL  PYROMETER  247 

where 

k  =  logeci  —  5  log  X. 

Let  EI  and  E%  be  the  energies  for  a  particular  wave  length  radi- 
ated at  the  temperatures  TI  and  T2,  respectively.  Substituting 
these  values  in  eq.  (9)  and  subtracting,  we  have 

1        1  \  xinN 

rJ 

If  T%  is  a  standard  temperature  and  TI  an  unknown,  then  by 
measuring  EI  and  E2  or  their  ratio,  by  appropriate  means,  TI 
may  be  computed.  Solving  eq.  (10)  for  TI  and  using  common 
logarithms, 

Tl  =  ^  J:_ 

X  2.303  lo° 


190.  The  Optical  Pyrometer.  —  One  of  the  most  convenient 
forms  of  the  optical  pyrometer  is  that  devised  by  Holborn  and 


B 
FIG.  130. — Holborn  and  Kurlbaum  optical  pyrometer. 

Kurlbaum.  It  consists  of  a  telescope  in  the  focal  plane  of  which 
is  mounted  a  small  six  volt  lamp  with  either  a  carbon  or  tungsten 
filament,  as  shown  in  Fig.  130.  When  the  telescope  is  focused 
on  the  furnace  and  the  filament  is  lighted,  there  is  seen,  on  looking 
into  it,  a  field  of  uniform  illumination  with  a  fine  line  extending 
across  it.  If  the  filament  is  hotter  than  the  furnace,  it  appears 
as  a  bright  line  across  a  dark  background;  but  if  the  furnace  is 
hotter,  there  is  seen  a  dark  line  across  a  bright  background.  If 
filament  and  furnace  are  at  the  same  temperature,  the  line  disap- 
pears and  the  field  is  uniform  throughout.  The  eye  is  very 
sensitive  to  differences  of  brightness  and  a  difference  of  two  degrees 
between  furnace  and  filament  may  easily  be  detected.  Current 
for  the  filament  is  furnished  by  a  storage  battery,  controlled  by  a 


248  ELECTRICITY  AND  MAGNETISM 

rheostat  and  measured  by  an  ammeter.  The  indications  of  the 
pyrometer  are  thus  in  terms  of  the  filament  current.  If  the 
furnace  is  held  in  turn  at  a  series  of  known  temperatures  and 
the  filament  currents  for  balance  obtained,  a  calibration  curve 
may  be  plotted  showing  temperature  as  a  function  of  current. 

A  number  of  improvements  in  the  original  form  of  the  Holborn- 
Kurlbaum  pyrometer  have  been  made  by  Mendenhall.1  One  of 
them  is  a  method  by  which  such  an  instrument  may  be  calibrated 
over  a  wide  range  using  only  one  standard  temperature.  This  is 
accomplished  by  holding  the  temperature  of  the  furnace  constant 
and  rotating  between  it  and  the  pyrometer  a  sectored  disk  which 
allows  only  a  known  fraction  of  the  energy  to  enter  the  telescope. 
This  is  equivalent  to  reducing  the  temperature  of  the  furnace. 
Suppose  the  fraction  of  the  energy  transmitted  is  R.  Then  EI 
=  RE2.  Substituting  this  value  in  eq.  (11),  we  have,  for  the 
apparent  temperature  of  the  furnace, 

Tl  =  ^  T  (12) 

X    ^  +  2.303  Iog10^ 

By  using  a  series  of  sectors,  for  example  with  R  equal  to  ^,^,^'s, 
etc.,  a  series  of  apparent  temperatures  are  obtained,  and  the 
filament  temperatures  corresponding  to  each  may  be  determined. 
This  gives  a  calibration  for  the  instrument  for  ranges  below  the 
standard  temperature  actually  maintained  in  the  furnace.  The 
necessary  narrow  wave  length  band  is  secured  by  mounting 
behind  the  eyepiece  a  disk  of  red  glass  of  special  quality.  The 
instrument  also  may  be  used  to  measure  temperatures  above 
that  of  the  standard  by  using  the  sectored  disk  when  taking 
observations  on  .the  unknown  temperature,  thus  reducing  it  to  an 
apparent  lower  temperature  within  the  calibration  range  just 
determined.  For  example,  if  an  unknown  temperature  is 
observed  through  a  sector  of  transmission  ratio  R  and  is  found  to 
be  the  same  as  the  standard  temperature  772  then  the  unknown 
temperature  is  obtained  from  eq.  (11)  by  putting  E2  =  REi  which 
gives 


c2  +  2.303 


In  a  similar  manner  a  calibration  curve  may  be  computed  for  a 
given  sector  extending  the  range  of  the  instrument  to  any  desired 
1  MENDENHALL,  Phys.  Rev.,  vol.  35,  1910,  p.  74. 


PHOTOMETER  AND  OPTICAL  PYROMETER 


249 


value.  For  this  purpose,  it  is  only  necessary  to  substitute  for  772 
in  eq.  (13)  the  value  of  temperature  corresponding  to  each  par- 
ticular current  read  off  from  the  original  calibration  curve.  These 
computed  values  of  T\  plotted  against  the  corresponding  values 
of  filament  current  give  the  calibration  curve  for  the  instrument 
when  used  with  the  sectors  to  measure  unknown  temperatures. 

It  should  be  borne-in  mincfthat  the  Wien  radiation  law  upon 
which  this  method  is  based  holds  only  for  black  body  radiation, 
and  that  the  method  of  calibration  just  described  makes  use  of  a 
black  body  as  a  source  of  radiation.  If  it  should  be  used  to 
determine  the  temperature  of  some  other  body  such  as  a  heated 
filament  or  strip  of  metal  not  within  an  enclosure,  its  indications 
will  be  the  temperature  of  a  black  body  which  would  emit  the 
same  amount  of  energy  at  the  particular  wave  length  used  in  the 
calibration.  Since  no  other  body  emits  more  energy  at  any 
wave  length  than  a  black  body  at  the  same  temperature,  and 
most  substances  emit  less  than  a  black  body,  the  reading  of  the 
optical  pyrometer  will,  in  general,  be  too  low.  The  reading 
obtained  is  called  the  " black  body  temperature."  Mendenhall 
and  Forsythe1  have  made  an  extended  study  of  the  differences 
between  the  " black  body"  and  "true"  temperatures  of  a  great 
many  substances  with  the  result  that  the  optical  pyrometer 
may  now  be  very  generally  used  to  determine  actual  tempera- 
tures. A  few  of  their  values  for  carbon  and  tungsten  are  given 
below : 


Black   body 
temperature 


Corresponding  true  temperature 


Degrees  Centigrade 

Tungsten,    Degrees    C. 

Carbon,  Degrees  C. 

1,000 

1,068 

1,012 

1,200 

1,273 

1,222 

1,400 

1,486 

1,430 

1,600 

1,700 

1,638 

1,800 

1,910 

1,847 

2,000 

2,126 

2,056 

2,200 

2,345 

2,400 

2,565 

2,600 

2,783 

2,700 

2,890 

1  MENDENHALL  and  FORSYTHE,  Astrophysical  Jour.}  vol.  37,  1913,  p.  389. 


250  ELECTRICITY  AND  MAGNETISM 

191.  Experiment  41. — Connect  the  apparatus  as  shown  in 
Fig.  130.  Ascertain  the  heating  currents  to  be  used  through  the 
two  windings  of  the  furnace  and  take  care  that  they  are  never 
exceeded,  particularly  through  the  inner  platinum  winding. 
Find  out,  also,  the  -maximum  current  allowable  for  the  filament 
of  the  pyrometer  lamp  Fill  the  vessel  containing  the  cold 
junction  of  the  thermocouple  with  cracked  ice  to  maintain  it  at 
0°  C.  Measure  the  E.M.F.  of  the  thermocouple  with  a  low  resis- 
tance potentiometer,  special  instructions  for  which  are  given  in 
chap.  IV.  A  calibration  curve  is  furnished  with  the  thermo- 
couple. Focus  the  eyepiece  of  the  telescope  on  the  lamp  fila- 
ment and  as  soon  a  the  furnace  is  warm  enough  to  permit 
it,  focus  the  telescope  so  that  the  inner  circle  of  the  furnace  is 
distinctly  seen  As  the  furnace  heats  up,  determine  its  tem- 
perature with  the  thermocouple  and  balance  the  pyrometer  every 
few  minutes. 

When  a  temperature  of  1,200°  C.  has  been  reached,  reduce  the 
heating  current  through  both  windings  and  allow  the  temperature 
to  rise  slowly  to  about  1,300°  C.  and  then  hold  the  furnace  con- 
stant at  this  value.  When  holding  the  temperature  constant,  it 
is  best  to  leave  the  potentiometer  setting  fixed  and  keep  the 
galvanometer  balanced  by  adjusting  the  heating  current  rheostat. 
When  a  steady  state  has  been  secured,  make  several  settings  of 
the  pyrometer.  Then  introduce  the  J^  sector  and  with  the  motor 
running,  again  make  several  settings  on  this  apparent  temperature. 
Repeat,  using  the  K>  Mo>  Mo>  Mo»  and  ^20  sectors.  Check  the 
settings  with  no  sector  between  each  replacement  to  insure  con- 
stancy of  conditions.  Two  observers  are  required  for  this 
experiment,  one  to  manipulate  the  pyrometer,  and  one  to  hold 
the  furnace  temperature  constant.  Measure  the  temperature 
of  the  filaments  of  several  incandescent  lamps  of  different  types 
and  candle  power  using  such  a  sector  that  the  current  through  the 
pyrometer  lamp  lies  within  the  range  covered  by  the  calibration. 

Report. — Compute  the  effective  temperatures  below  the  stand- 
ard temperature  secured  by  the  various  sectors  by  use  of  eq.  (12). 
Use  for  02  the  value  14,350  and  for  the  wave  length  0.658. 
The  standard  temperature  is  that  in  degrees  absolute  at  which 
the  furnace  was  held  constant  and  is  obtained  from  the  calibra- 
tion curve  for  the  thermocouple.  The  computed  values  of  TI 
are  also  in  degrees  absolute.  Plot  temperatures  below  that  of  the 
furnace. 


PHOTOMETER  AND  OPTICAL  PYROMETER  251 

Plot  calibration  curves  for  values  above  that  of  the  furnace  for 
the  Ko>  ^1*0>  and  K20  sectors,  by  use  of  eq.  13.  To  do  this, 
read  from  the  first  curve  the  values  of  T  for  a  series  of  values  of 
filament  currents.  Substitute  these  values  of  T^  in  eq.  13  using 
for  R  the  appropriate  ratio ^  These  values  of  T7,  when  plotted 
against  the  currents,  give  the  calibration  curve  for  a  given 
sector  for  the  high  *ange. 

Read  from  these  curves  the  black  body  temperatures  of  the 
lamps  measured  and  by  use  of  the  tables  given  above,  determine 
their  true  temperatures  in  degrees  centigrade. 

If  the  temperature  of  the  sun  is  about  6,000°  C.,  find  the  size 
of  sector  opening  necessary  to  measure  it  on  the  instrument  used. 


APPENDIX 
CALCULATION  OF  INDUCTANCE  AND  CAPACITANCE 

In  designing  electrical  apparatus  and  in  checking  the  results  of 
bridge  measurements  it  is  often  advantageous  to  determine  the 
inductance  of  coils  by  calculation  from  their  dimensions  and 
number  of  turns.  In  connection  with  its  work  in  establishing 
primary  units  of  inductance,  the  United  States  Bureau  of  Stand- 
ards made  an  exhaustive  study  of  the  formulas  for  this  purpose, 
and,  besides  extending  those  available  at  the  time,  developed  a 
number  of  new  ones.  A  comprehensive  collection  of  inductance 
formulas,  together  with  numerical  examples,  is  given  in  the 
Bulletin  of  the  Bureau  of  Standards,  vol.  8,  1912,  pp.  1  to  237. 
This  publication  is  known  also  as  Scientific  Paper  169.  In 
another  publication,  " Radio  Instruments  and  Measurements," 
Circular  74,  there  is  given  a  series  of  simplified  formulas  which 
yield  results  accurate  to  one-tenth  of  one  per  cent. 

Three  formulas,  taken  from  Circular  74,  are  given  below.  They 
apply  to  the  coils  most  commonly  used  in  every  day  practice. 
Lengths  and  other  dimensions  are  expressed  in  centimeters, 
and  the  inductance  calculated  is  given  in  microhenries.  One 
henry  =  103  millihenries  =  106  microhenries  =  109  C.G.S. 
electromagnetic  units.  It  is  assumed  that  the  coil  is  placed  in 
air  or  other  medium  whose  permeability  is  unity,  and  that  no 
iron  is  in  the  vicinity. 

/.  Single  Layer  Coil  or  Solenoid. — Nagaoka's  Formula. 

L  _  0.039480^  (1) 

where  n  =  number  of  turns  of  coil 

a  =  radius  of  coil,  i.e.,  axis  to  center  of  any  wire 

b  =  length  of  coil,  i.e.,  number  of  turns  times  distance 

between  centers  of  adjacent  turns. 
K  is  a  correction  factor  made  necessary  by  the  demagnetizing 

action  of  the  ends  of  the  coil  and  is  a  function  of  -r-  •    Its  value 

b 

may  be  read  from  Table  I.     If  the  coil  is  very  long  compared  to 

252 


INDUCTANCE  AND  CAPACITANCE 


253 


its  diameter,  K  =  1.  Formula  (1)  takes  no  account  of  the  size 
or  shape  of  the  cross-section  of  the  wire  and  assumes  that  the 
diameter  of  the  wire  is  small  compared  to  the  dimensions  of  the 
coil,  and  that  the  coil  is  compactly  wound. 

TABLE  I. — VALUES  OF  K  FOR  USE  IN  FORMULA  I 


Diameter 
Length 

K 

Differ- 
ence 

P 
Diameter 
Length 

K 

Differ- 
ence 

Diameter 
Length 

K 

Differ- 
ence 

0.00 

I  .  0000 

-0.0209 

2.00 

0.5255 

-0.0118 

7.00 

0.2584 

-0.0047 

.05 

.9791 

203 

2.10 

.5137 

112 

7.20 

.2537 

45 

.10 

.9588 

197 

2.20 

.  5025 

107 

7.40 

.2491 

43 

.15 

.9391 

190 

2.30 

.4918 

102 

7.60 

.2448 

42 

.20 

.9201 

185 

2.40 

.4816 

97 

7.80 

.2406 

40 

0.25 

0.9016 

-0.0178 

2.50 

0.4719 

-0.0093 

8.00 

0.2366 

-0.0094 

.30 

.8838 

173 

2.60 

.4626 

89 

8.50 

.2272 

86 

.35 

.8665 

167 

2.70 

.4537 

85 

9.00 

.2185 

79 

.40 

.8499 

162 

2.80 

.4452 

82 

9.50 

.2106 

73 

.45 

.8337 

156 

2.90 

.4370 

78 

10.00 

.  2033 

0.50 

0.8181 

-0.0150 

3.00 

0.4292 

-0.0075 

10.0 

0.2033 

-0.0133 

.55 

.8031 

146 

3.10 

.4217 

72 

11:0 

.1903 

113 

.60 

.7885 

140 

3.20 

.4145 

70 

12.0 

.1790 

98 

.65 

.7745 

136 

3.30 

.4075 

67 

13.0 

.1692 

87 

.70 

.7609 

131 

3.40 

.4008 

64 

14.0 

.  1605 

78 

0.75 

0.7478 

-0.0127 

3.50 

0  .  3944 

-0.0062 

15.0 

0.1527 

-0.0070 

.80 

.7351 

123 

3.60 

.3882 

60 

16.0 

.1457 

63 

.85 

.7228 

118 

3.70 

.3822 

58 

17.0 

.1394 

58 

.90 

.7110 

115 

3.80 

.3764 

56 

18.0 

.1336 

52 

.95 

.6995 

111 

3.90 

.3708 

54 

19.0 

.1284 

48 

1.00 

0.6884 

-0.0107 

4.00 

0.3654 

-0.0052 

20.0 

0.1236 

-0.0085 

1.05 

.6777 

104 

4.10 

.  3602 

51 

22.0 

.1151 

73 

1.10 

.6673 

100 

4.20 

.3551 

49 

24.0 

.1078 

83 

1.15 

.  6573 

98 

4.30 

.3502 

47 

26.0 

.1015 

56 

1.20 

.  6475 

94 

4.40 

.  3455 

46 

28.0 

.0959 

49 

1.25 

0.6381 

-0.0091 

4.50 

0.3409 

-0.0045 

30.0 

0.0910 

-0.0102 

1.30 

.6290 

89 

4.60 

.3364 

43 

35.0 

.  0808 

80 

1.35 

.6201 

86 

4.70 

.3321 

42 

40.0 

.0728 

64 

1.40 

.6115 

84 

4.80 

.3279 

41 

45.0 

.0664 

53 

1.45 

.6031 

81 

4.90 

.3238 

40 

50.0 

.0611 

43 

1.50 

0.5950 

-0.0079 

5.00 

0.3198 

-0.0076 

60.0 

0.0528 

-0.0061 

1.55 

.5871 

76 

5.20 

.3122 

72 

70.0 

.0467 

48 

1.60 

.5795 

74 

5.40 

.3050 

69 

80.0 

.0419 

38 

1.65 

.5721 

72 

5.60 

.2981 

65 

90.0 

.0381 

31 

1.70 

.5649 

70 

5.80 

.2916 

62 

100.0 

.  0350 

1.75 

0.5579 

-0.0068 

6.00 

0.2854 

-0.0059 

1.80 

.5511 

67 

6.20 

.2795 

56 

1.85 

.5444 

65 

6.40- 

.2739 

54 

1.90 

.5379 

63 

6.60 

.  2685 

52 

1.95 

.5316 

61 

6.80 

.2633 

49 

254 


ELECTRICITY  AND  MAGNETISM 


II.  Multiple  Layer  Coil. — For  a  long  coil  with  few  layers,  the 
inductance  is  given  by 

0.01257n2ac 
L  =  Ls  -         — r —  —  (U.o9o  -h  &g)  (2) 

where  Ls  =  inductance  of  mean  single  layer  given  by  formula  (1) 
n  =  number  of  turns  of  the  coil 
a  =  radius  of  coil  measured  from  the  axis  to  the  center 

of  cross-section  of  the  winding 
b  =  length  of  coil  =  distance  between  centers  of  turns 

times  number  of  turns  in  one  layer 
c  =  radial  depth  of  winding  =  distance  between  centers 

of  two  adjacent  layers  times  the  number  of  layers. 

Bs  =  correction  given  in  Table  II  in  terms  of  the  ratio  - ' 
TABLE  II. — VALUES  OF  Bs  FOR  USE  IN  FORMULA  II 


b/c 

Bs 

b/c       Bs 

\ 

b/c 

Bs 

1 

0.0000 

11 

0.2844      21 

0.3116 

2 

0.1202 

12 

0.2888      22 

0.3131 

3 

0.1753 

13 

0.2927 

23 

0.3145 

4 

0.2076 

14 

0.2961 

24 

0.3157 

5 

0.2292 

15 

0.2991 

25 

0.3169 

6 

0.2446      16 

0.3017 

26 

0.3180 

7 

0.2563      17 

0.3041 

27 

0.3190 

8 

0.2656      18 

0.3062 

28 

0.3200 

9 

0.2730      19 

0.3082 

29      0.3209 

10 

0.2792 

20 

0.3099 

30      0.3218 

///.  Short  Circular  Coil  with  Rectangular  Cross  Section. — For 
a  coil  having  a  shape  such  as  shown  in  Fig.  131,  the  inductance  is 
given  by  a  formula  due  to  Stefan.  It  is  deduced  on  the  assump- 
tion that  the  wire  is  rectangular  in  cross-section,  and  that  the 
insulating  space  between  turns  is  negligible.  Further,  the  axial 
and  radial  dimensions  of  the  winding  are  supposed  to  be  small 
compared  to  the  mean  radius  of  the  coil. 
Let  a  =  the  mean  radius  of  the  winding  measured  from  the  axis 

to  the  center  of  the  cross-section 
b  =  the  axial  dimension  of  the  cross-section 
c  =  the  radial  dimension  of  the  cross-section 
d  —  \/b2  +  c2  =  the  diagonal  of  the  cross-section 
n  =  number  of  turns  of  rectangular  wire. 


INDUCTANCE  AND  CAPACITAMCE 


255 


There  are  two  cases  depending  upon  the  relative  values  of  6 
and  c. 

Case  1.     b>c. 

L  =  0.01257an2[2.303^r+ 

~< 


Case  2.     6<c. 


L  =  0.01257cm*  [2.303( 


T 

J 


FIG.  131. — Multiple  layer  coil  with  winding  of  rectangular  cross  section. 


The  constants  yi9  yz,  and  2/3  depend  upon  relative  values  of 
b  and  c,  and  are  given  in  Table  III.  The  ratio  of  these  quantities 
is  always  to  be  taken  so  as  to  give  a  proper  fraction;  i.e.,  in  for- 
mula (3),  use  c/6,  and  in  formula  (4),  use  6/c.  In  eq.  (3),  y\  is  the 
same  function  of  c/b  that  it  is  of  b/c  in  eq.  (4). 


256  ELECTRICITY  AND  MAGNETISM 

TABLE  III. — CONSTANTS  USED  IN  FORMULAS  (3)  and  (4) 


b/c  or  c/b 

yi 

Differ- 
ence 

c/b 

2/2 

Differ- 
ence 

b/c 

2/3 

Differ- 
ence 

\ 

0 

0.5000 

0.0253 

0 

0.125 

0.002 

0 

0.597 

0.002 

0.025 

.5253 

237 

.05 

.5490 

434 

0.05 

.127 

5 

0.05 

.599 

3 

.10 

.5924 

386 

.10 

.132 

10 

.10 

.602 

6 

0.15 

0.6310 

0.0342 

0.15 

0.142 

0.013 

•  0.15 

0.608 

0.007 

.20 

.6652 

301 

.20 

.155 

16 

.20 

.615 

9 

.25 

.6953 

266 

.25 

.171 

20 

.25 

.624 

9 

.30 

.7217 

230 

.30 

.192 

23 

.30 

.633 

10 

0.35 

0.7447 

0.0198 

0.35 

0.215 

0.027 

0.35 

0.643 

0.011 

.40 

.7645 

171 

.40 

.242 

31 

.40 

.654 

11 

.45 

.7816 

144 

.45 

.273 

34 

.45 

.665 

12 

.50 

.7960 

121 

.50 

.307 

37 

.50 

.677 

13 

0.55 

0.8081 

0.0101 

0.55 

0.344 

0.040 

0.55 

0.690 

0.012 

.60 

.8182 

83 

.60 

.384 

43 

.60 

.702 

13 

.65 

.8265 

66 

.65 

.427 

47 

.65 

.715 

14 

.70 

.8331 

52 

.70 

.474 

49 

.70 

.729 

13 

0.75 

0  .  8383 

0  .  0039 

0.75 

0.523 

0.053 

0.75 

0.742 

0.014 

.80 

.8422 

29 

.80 

.576 

56 

.80 

.756 

15 

.85 

.8451 

19 

.85 

.632 

59 

.85 

.771 

15 

.90 

.8470 

10 

.90 

.690 

62 

.90 

.786 

15 

0.95 

0.8480 

0.0003 

0.95 

0.752 

0.064 

0.95 

0.801 

0.015 

1.00 

.8483 

1.00 

.816 

1.00 

.816 

IV.  Coil  of  Round  Wire  Wound  in  a  Channel  of  Rectangular 
Cross-section.  If  the  insulation  is  not  too  thick,  eqs.  (3)  and  (4) 
give  a  very  close  approximation  for  the  case  in  which  ordinary 
round  wire  is  used.  When  the  percentage  of  the  cross-section 
occupied  by  the  insulating  space  is  large,  the  following  correction 
must  be  added  to  these  formulas. 


AL  =  0.01257cm2  [2.303  logio  ?  +  0.1551 

(IQ  J 

where  D  =  distance  between  centers  of  adjacent  wires 
do  =  diameter  of  the  bare  wire. 


(5) 


CALCULATION  OF  CAPACITANCE 

The  following  formulas1  may  be  used  to  calculate  the  capac- 
itance of  condensers  of  the  common  forms.  The  dimensions  of 
the  condensers  are  measured  in  centimeters,  and  the  capacitance 
is  given  in  micro-microfarads.  In  these  formulas,,  no  correction 

1  Cir.  74,  U.  S.  Bureau  of  Standards,  p.  235. 


INDUCTANCE  AND  CAPACITANCE  257 

is  made  for  the  curving  of  the  electrostatic  field  at  the  edges  of 
plates,  etc.,  and  it  is  assumed  that  the  distance  between  plates  is 
small  compared  to  their  linear  dimensions. 
V.  Parallel  Plate  Condenser 

C  '=0.0885*4  (6) 

As  •*• 

where  S  =  surface  area  of  one  plate 
T  =  thickness  of  dielectric 

K  =   dielectric  constant  (K  =  1  for  air,  and  for  most  sub- 
stances, lies  between  1  and  10). 

If,  instead  of  a  single  pair  of  plates,  there  are  N  similar  plates 
with  dielectric  between  them  alternate  plates  being  connected 
in  parallel, 


C  =  0.0885#  (7) 

VI.  Variable  Condenser  with  Semi-circular  Plates 

C  -  0.1390K^--^1^!)  (8) 

where  N  =  total  number  of  plates 

TI  =  outside  radius  of  the  plates 
r2  =  inside  radius  of  the  plates 
T  =  thickness  of  dielectric 
K  =  dielectric  constant 

This  formula  gives  the  maximum  capacitance,  i.e.,  when  the 
movable  plates  are  entirely  within  the  spaces  between  the  fixed 
plates.  As  the  movable  plates  are  rotated  out,  the  capacitance 
decreases  in  direct  proportion  to  the  angle  through  which  they 
are  turned. 

VII.  Isolated  Disk  of  Negligible  Thickness 

C  =  0.354d  (9) 

where  d  =  diameter  of  the  disk 

VIII.  Isolated  Sphere 

C  =  0.556d  (10) 

where  d  =  diameter  of  the  sphere 

IX.  Two  Concentric  Spheres 


C  =  1.112K  (11) 

ri  -  rz 

17 


258  ELECTRICITY  AND  MAGNETISM 

where  n  =  inner  radius  of  outside  sphere 
r2  =  outer  radius  of  inner  sphere 
K  =  dielectric  constant  of  material  between  spheres. 

X.   Two  Coaxial  Cylinders 

0  =  0.2416*--'- 
log,.  - 

where  /  =  length  of  each  cylinder 

ri  =  inner  radius  of  outer  cylinder 
r2  =  outer  radius  of  inner  cylinder 
K  =  dielectric  constant  of  material  between  cylinders. 


INDEX 


Alpha  rays,  212 

Alternating    current    galvanometer, 

165 

Ammeter,  74 
adjustment  of,  76 
calibration  of,  80 
Ampere  turn,  definition  of,  102 
Amplification  factor  of  electron  tube, 

227 

dynamic  method  for,  231 
Anderson     modification     of     Ma'x- 

well's  bridge,  129 
Atom,  structure  of,  200 
Audio-oscillator,  155 

B 

Ballantine,     dynamic     method    for 
resistance  of  electron  tube,  234 
Battery  test,  53 
Beta  rays,  213 
Black  body,  244 

temperatures    corrected    to    true, 
249 


Campbell,    measurement   of   induc- 
tance, 181 
Carey-Foster  bridge  for   resistance, 

42 
method    for    mutual    inductance, 

124 
Cathode  glow,  205 

rays,  208 

Checking  devices  for  ballistic  gal- 
vanometer, 33 
Comparison  of  cells,  63 
Condensers,  capacitance  of,  86 
comparison  of,  89 
grouping  of,  87 

measurement    by    Fleming    and 
Clinton  commutator,  93 


Crooke's  dark  space,  205 
Current,    measured    by    electrody- 
namometer,  72 

Kelvin  balance,  70 

potentiometer,  79 

D 

Damped  sine  wave,  135 

Decrement,  logarithmic,  136 

Demagnetizing  factor,  96 

Discharge    of    condenser,    aperiodic 

discharge,  131 

critically  damped  discharge,  132 
oscillatory  discharge,  133 
through  gases,  theory  of,  205 

Duddell  therm o-galvanometer,  162 


Effective    value    of    an    alternating 

current,  148 

Electrodynamometer,  Siemens,  72 
Electrolytes,  resistance  of,  194 
Electrons,  198 
Electron  tubes,  218 

amplification  factor  of,  227 

as  oscillator,  159 

characteristics  for,  228 

impedance  of,  234 


Faraday  dark  space,  206 

Fleming  and  Clinton  commutator, 
92 

Fluxmeter,  Grassot,  31 

Forsythe  and  Mendenhall,  correc- 
tions for  black  body  tempera- 
tures, 249 

Frequency  bridge,  186 

G 

Galvanometer,  description  of,  17 
ballistic  galvanometer,  theory  of, 
25 


259 


260 


INDEX 


Galvanometer,  constant  of,  24 
current  galvanometer,  19 
D'Arsonval  galvanometer,  18 
figure  of  merit,  22 
Thomson  galvanometer,  17 

Gamma  rays,  214 

Gauss,  definition  of,  101 

Gilbert,  definition  of,  101 

Graham,  potential  gradient  in  dis- 
charge tubes,  206 

Grover,  phase  angle  of  condensers, 
191 


II 


Heaviside's  bridge   for  inductance, 

180 
Heydweiller's   network    for   mutual 

inductance,  177 
Holborn    and    Kurlbaum's    optical 

pyrometer,  247 

Hopkinson's  bar  and  yoke,  107 
Hysteresis,  104 

measurement  of,  114 


I 


Impedance,  139 
Inductance,  117 

calculation  of,  252 

coefficients  of,  118 

comparisons  of,  120 

standards  of,  119 
Induction,  magnetic,  96 
Insulation  resistance,  measurement 

of,  46 

Intensity  of  magnetization,  96 
Internal  resistance  of  cells,  50 

condenser     and    ballistic     galva- 
nometer method  for,  52 

voltmeter  ammeter  method  for,  51 
lonization,  theory  of,  151 


K 


Kaufmann,    variation    of      -    with 

m 

velocity,  214 
Kelvin  current  balance,  70 


Kelvin   current  balance,  70  double 
bridge  for   resistance  measure- 
ment, 40 
galvanometer,  17 

Kennelly  and  Pierce,  motional  im- 
pedance, 190 

Kenotron,  224 

Keys,  2 

Kumagen,  155 

Kurlbaum,  optical  pyrometer,  247 


Lummer-Brodhun  photometer,  241 
M 

Magnetic  circuit,  99 
shields,  18 

Magnetism,  general  principles,  94 

Magnetization  curves,  103 

Maxwell,  definition  of,  101 

Maxwell's  bridge  for  mutual  induc- 
tance, 169 

Mendenhall,  use  of  optical  pyrom- 
eter, 248 

correction   for   black   body    tem- 
peratures, 249 

Microphone  hummer,  154 

Miller,  amplification  factor  of  elec- 
tron tubes,  230 

Motional  impedance,  191 

Motor  generator,  153 

Multipliers  for  voltmeter,  77 

Mutual  inductance  bridge,  184 

N 

Nagaoka's  inductance  formula,  252 
Negative  glow,  206 
Notebooks,  6 


Oersted,  definition  of,  102 
Ohm's  law,  35 


Permeability,  magnetic,  97 
Phase  angle  of  condensers,  191 
Photo-electric  effect,  219 
Photometer,  240 


INDEX 


261 


Pierce  and  Kennelly,  motional  im- 
pedance, 190 
Pohl's  commutator,  2 
Polarization  of  cell,  54 
Positive  column,  206 
Post-office  box,  38 
Potentiometer,  description  of,  95 

Leeds  and  Northrup,  58  & 

Tinsley,  62 

Wolff,  60 
Power,  measurement  of,  82 

factor,  definition  of,  150 

of  condensers,  191 
Pyrometer,  optical,  243 


R 


Radiation,  intensity  of,  239 
Radioactive  substances,  212 
Reactance,  139 
Resistance,  specific,  35 

measurement  of  high  resistance, 

46 

of  low  resistance,  40 
temperature  coefficient  of,  36 
Resistances    for    current    measure- 
ments, 80 

Resonance,  electrical,  142 
parallel  resonance,  144 
series  resonance,  143 
Rheostats,  3 

Root  mean  square  value  of  an  alter- 
nating current,  149 
Rowland  ring,  111 


S 


Saturation  current,  201 
Sechometer,  151 

Siemen's  electrodynamometer,  72 
Sine  wave,  vector  representation  of, 

141 
Space  charge,  222 

current,  221 
Specific  resistance,  35 
Spectrum,  distribution  of  energy  in, 

245 
Standard  cell,  E.M.F.  of,  11 

temperature  coefficient  of,  12 


Stefan  Boltzman  law,  245 
Steinmetz  coefficient,  106 
Stroude  and  Gate's  bridge,  173 
Susceptibility,  97 
Switchboard,  5 
Switches,  2 


Telephone  receiver,  161 
Temperature    coefficient    of    resis- 
tance, 36 

standard  cell,  12 

Thermo-galvanometer,  Duddell,  162 

Time  constant  of  circuit  containing 

resistance  and  inductance,  126 

and  capacitance,  128 

inductance  and  capacitance,  131 
Tinsley  potentiometer,  62 
Trowbridge's  bridge,  174 
Tungar  rectifier,  235 

U 

Units,  systems  of,  7 
electromagnetic,  8 
electrostatic,  8 
practical,  9 

rationalized  .practical,  13 
ratios  of,  12 


Variable  impedance  circuit,  188 
Vibration  galvanometer,  163 
Volt  box,  67 
Voltmeter,  adjustment  of,  74 

calibration  of,  68 
Vreeland  oscillator,  157 

W 

Wattmeters,  description  of,  82 

compensation  of,  83 

calibration  of,  84 

Wein,  phase  angle  of  condensers,  191 
Wein's  law,  246 
Weston  instruments,  75 

standard  cell,  64 
Wheatstone  bridge,  36 
Wire  interrupter,  153 
Wolff  potentiometer,  60 


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m-*m     -g^r  »*  n  p  •  ?  j  ?•    ".  

DEC  **  1947 

JAN  1  o  1943 

MAK  Z  2  1846 

fif*T        A    1Q/IQ 

Ms  vm 

NUV  i        J 

fl  JAN  5     itfi3 

"  •• 

10m-7,'44(1064s) 

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Engineering 
Library 


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